In [64]:

    

%run "../src/start_session.py"
%run "../src/recurrences.py"


    

In [50]:

    

import oeis


    

In [3]:

    

s = oeis.oeis_search(id=48993)


    

    




*

In [4]:

    

s()


    

    
Out[4]:




Results for query: https://oeis.org/search?q=id%3AA048993&start=0&fmt=json
A048993: Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0<=k<=n.

by N. J. A. Sloane, Dec 11 1999
Keywords: nonn,tabl,nice
Data:
$$
\begin{array}{c|ccccccccccc}
n, k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
0 & 1 &  &  &  &  &  &  &  &  & \\1 & 0 & 1 &  &  &  &  &  &  &  & \\2 & 0 & 1 & 1 &  &  &  &  &  &  & \\3 & 0 & 1 & 3 & 1 &  &  &  &  &  & \\4 & 0 & 1 & 7 & 6 & 1 &  &  &  &  & \\5 & 0 & 1 & 15 & 25 & 10 & 1 &  &  &  & \\6 & 0 & 1 & 31 & 90 & 65 & 15 & 1 &  &  & \\7 & 0 & 1 & 63 & 301 & 350 & 140 & 21 & 1 &  & \\8 & 0 & 1 & 127 & 966 & 1701 & 1050 & 266 & 28 & 1 & \\9 & 0 & 1 & 255 & 3025 & 7770 & 6951 & 2646 & 462 & 36 & 1
\end{array}
$$
Comments:

Also known as Stirling set numbers.
S(n,k) enumerates partitions of an n-set into k nonempty subsets.
The o.g.f. for the sequence of diagonal k (k=0 for the main diagonal) is G(k,x)= ((x^k)/(1-x)^(2k+1))sum(A008517(k,m+1)*x^m,m=0..k-1). A008517 is the second-order Eulerian triangle. - Wolfdieter Lang, Oct 14 2005.
From Philippe Deléham, Nov 14 2007: 

Sum_{k, 0<=k<=n}S(n,k)*x^k = B_n(x), where B_n(x) = Bell polynomials. The first few Bell polynomials are:
B_0(x) = 1;
B_1(x) = 0 + x;
B_2(x) = 0 + x + x^2;
B_3(x) = 0 + x + 3x^2 + x^3;
B_4(x) = 0 + x + 7x^2 + 6x^3 + x^4;
B_5(x) = 0 + x + 15x^2 + 25x^3 + 10x^4 + x^5;
B_6(x) = 0 + x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6;


This is the Sheffer triangle (1, exp(x) - 1), an exponential (binomial) convolution triangle. The a-sequence is given by A006232/A006233 (Cauchy sequence). The z-sequence is the zero sequence. See the link under A006232 for the definition and use of these sequences. The row sums give A000110 (Bell), and the alternating row sums give A000587 (see the Philippe Deléham formulas and crossreferences below). - Wolfdieter Lang, Oct 16 2014
Also the inverse Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

Formulae:

S(n, k) = k*S(n-1, k) + S(n-1, k-1), n>0; S(0, k) = 0, k>0; S(0, 0)=1.
Equals [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ..] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deléham's operator defined in A084938.
Sum_{k = 0..n} x^k*S(n, k) = A213170(n), A000587(n), A000007(n), A000110(n), A001861(n), A027710(n), A078944(n), A144180(n), A144223(n), A144263(n) respectively for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7. - Philippe Deléham, May 09 2004, Feb 16 2013
S(n, k) = sum{i=0..k, (-1)^(k+i)binomial(k, i)i^n/k!}. - Paul Barry, Aug 05 2004
Sum(k*S(n,k), k=0..n)=B(n+1)-B(n), where B(q) are the Bell numbers (A000110). - Emeric Deutsch, Nov 01 2006
Equals the inverse binomial transform of A008277. - Gary W. Adamson, Jan 29 2008
G.f.: 1/(1-xy/(1-x/(1-xy/(1-2x/(1-xy/1-3x/(1-xy/(1-4x/(1-xy/(1-5x/(1-... (continued fraction equivalent to Deleham DELTA construction). - Paul Barry, Dec 06 2009
G.f.: 1/Q(0), where Q(k) = 1 -(y+k)x - (k+1)y*x^2/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 09 2013
Inverse of padded A008275 (padded just as A048993 = padded A008277). - Tom Copeland, Apr 25 2014
E.g.f. for the row polynomials s(n,x) = sum(S(n,k)x^k, k=0..n) is exp(x(exp(z)-1)) (Sheffer property). E.g.f. for the k-th column sequence with k leading zeros is ((exp(x)-1)^k)/k! (Sheffer property). - Wolfdieter Lang, Oct 16 2014

Cross references:

See especially A008277 which is the main entry for this triangle.
Cf. A008275, A039810-A039813, A048994.
A000110(n) = sum(S(n, k)) k=0..n, n >= 0. Cf. A085693.
Cf. A084938, A106800 (mirror image), A213061.

Links:

David W. Wilson, <a href="/A048993/b048993.txt">Table of n, a(n) for n = 0..10010</a>
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.
P. Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
R. M. Dickau, Stirling numbers of the second kind
G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.
FindStat - Combinatorial Statistic Finder, The number of blocks in the set partition.
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
C. M. Ringel, The Catalan combinatorics of the hereditary artin algebras, arXiv preprint arXiv:1502.06553, 2015
X.-T. Su, D.-Y. Yang, W.-W. Zhang, A note on the generalized factorial, Australasian Journal of Combinatorics, Volume 56 (2013), Pages 133-137.

References:

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.
J. H. Conway and R. K. Guy, The Book of Numbers, Springer, p. 92.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 244.
J. Riordan, An Introduction to Combinatorial Analysis, p. 48.

In [3]:

    

from sympy.functions.combinatorial.numbers import stirling


    

In [76]:

    

s = IndexedBase('s')
n, k = symbols('n k')

stirling_recurrence_spec = recurrence_spec(recurrence_eq=Eq(s[n+1, k+1], s[n, k] + (k+1)*s[n, k+1]), 
                                           recurrence_symbol=s, 
                                           variables=[n,k])


    

In [77]:

    

stirling_recurrence_spec


    

    
Out[77]:




$\left(\Theta, \Gamma\right)_{n,k}^{s}$ where: 
$\Theta = \left\{ s_{n + 1,k + 1} = \left(k + 1\right) s_{n,k + 1} + s_{n,k} \right\}$
$\Gamma = \left\{\begin{array}{c}\end{array}\right\}$

In [78]:

    

unfolded = stirling_recurrence_spec.unfold(depth=1)


    

In [79]:

    

unfolded


    

    
Out[79]:




$\left(\Theta, \Gamma\right)_{n,k}^{s}$ where: 
$\Theta = \left\{ s_{n + 1,k + 1} = k \left(\left(k + 1\right) s_{n - 1,k + 1} + s_{n - 1,k}\right) + k s_{n - 1,k} + \left(k + 1\right) s_{n - 1,k + 1} + s_{n - 1,k} + s_{n - 1,k - 1} \right\}$
$\Gamma = \left\{\begin{array}{c}s_{n,k} = k s_{n - 1,k} + s_{n - 1,k - 1}\\s_{n,k + 1} = \left(k + 1\right) s_{n - 1,k + 1} + s_{n - 1,k}\end{array}\right\}$

In [81]:

    

instantiated = unfolded.instantiate(strategy=raw(substitutions={n:9,k:4}))
instantiated


    

    
Out[81]:




$\left(\Theta, \Gamma\right)_{n,k}^{s}$ where: 
$\Theta = \left\{ s_{10,5} = s_{8,3} + 9 s_{8,4} + 25 s_{8,5} \right\}$
$\Gamma = \left\{\begin{array}{c}s_{9,4} = s_{8,3} + 4 s_{8,4}\\s_{9,5} = s_{8,4} + 5 s_{8,5}\end{array}\right\}$

In [58]:

    

known_binomials = {s[n,k]:stirling(n,k) for n in [10,9,8] for k in range(2,6)}

checked = instantiated.instantiate(strategy=raw(substitutions=known_binomials))
checked


    

    
Out[58]:




$\left(\Theta, \Gamma\right)_{n,k}^{s}$ where: 
$\Theta = \left\{ \mathrm{True} \right\}$
$\Gamma = \left\{\begin{array}{c}\end{array}\right\}$

In [59]:

    

checked.subsume()


    

    
Out[59]:




$\left(\Theta, \Gamma\right)_{n,k}^{s}$ where: 
$\Theta = \left\{ \mathrm{True} \right\}$
$\Gamma = \left\{\begin{array}{c}\end{array}\right\}$

In [82]:

    

based_recurrence_spec = unfolded.instantiate(strategy=based(arity=doubly_indexed()))
based_recurrence_spec


    

    
Out[82]:




$\left(\Theta, \Gamma\right)_{n,k}^{s}$ where: 
$\Theta = \left\{ s_{4,2} = s_{2,0} + 3 s_{2,1} + 4 s_{2,2} \right\}$
$\Gamma = \left\{\begin{array}{c}s_{3,1} = s_{2,0} + s_{2,1}\\s_{3,2} = s_{2,1} + 2 s_{2,2}\end{array}\right\}$

In [83]:

    

based_recurrence_spec.subsume()


    

    
Out[83]:




$\left(\Theta, \Gamma\right)_{n,k}^{s}$ where: 
$\Theta = \left\{ s_{4,2} = s_{2,0} + 3 s_{2,1} + 4 s_{2,2} \right\}$
$\Gamma = \left\{\begin{array}{c}s_{3,1} = s_{2,0} + s_{2,1}\\s_{3,2} = s_{2,1} + 2 s_{2,2}\end{array}\right\}$

In [66]:

    

ipython_latex_description(rec_spec=stirling_recurrence_spec, depths=range(9), based_instantiation=False,
                          arity=doubly_indexed())


    

    
Out[66]:





\begin{array}{c}s_{n + 1,k + 1} = k s_{n,k + 1} + s_{n,k} + s_{n,k + 1}\\
s_{n + 1,k + 1} = k^{2} s_{n - 1,k + 1} + 2 k s_{n - 1,k} + 2 k s_{n - 1,k + 1} + s_{n - 1,k} + s_{n - 1,k - 1} + s_{n - 1,k + 1}\\
s_{n + 1,k + 1} = k^{3} s_{n - 2,k + 1} + 3 k^{2} s_{n - 2,k} + 3 k^{2} s_{n - 2,k + 1} + 3 k s_{n - 2,k} + 3 k s_{n - 2,k - 1} + 3 k s_{n - 2,k + 1} + s_{n - 2,k} + s_{n - 2,k - 2} + s_{n - 2,k + 1}\\
s_{n + 1,k + 1} = k^{4} s_{n - 3,k + 1} + 4 k^{3} s_{n - 3,k} + 4 k^{3} s_{n - 3,k + 1} + 6 k^{2} s_{n - 3,k} + 6 k^{2} s_{n - 3,k - 1} + 6 k^{2} s_{n - 3,k + 1} + 4 k s_{n - 3,k} + 4 k s_{n - 3,k - 2} + 4 k s_{n - 3,k + 1} + s_{n - 3,k} + s_{n - 3,k - 3} - 2 s_{n - 3,k - 2} + s_{n - 3,k - 1} + s_{n - 3,k + 1}\\
s_{n + 1,k + 1} = k^{5} s_{n - 4,k + 1} + 5 k^{4} s_{n - 4,k} + 5 k^{4} s_{n - 4,k + 1} + 10 k^{3} s_{n - 4,k} + 10 k^{3} s_{n - 4,k - 1} + 10 k^{3} s_{n - 4,k + 1} + 10 k^{2} s_{n - 4,k} + 10 k^{2} s_{n - 4,k - 2} + 10 k^{2} s_{n - 4,k + 1} + 5 k s_{n - 4,k} + 5 k s_{n - 4,k - 3} - 10 k s_{n - 4,k - 2} + 5 k s_{n - 4,k - 1} + 5 k s_{n - 4,k + 1} + s_{n - 4,k} + s_{n - 4,k - 4} - 5 s_{n - 4,k - 3} + 5 s_{n - 4,k - 2} + s_{n - 4,k + 1}\\
s_{n + 1,k + 1} = k^{6} s_{n - 5,k + 1} + 6 k^{5} s_{n - 5,k} + 6 k^{5} s_{n - 5,k + 1} + 15 k^{4} s_{n - 5,k} + 15 k^{4} s_{n - 5,k - 1} + 15 k^{4} s_{n - 5,k + 1} + 20 k^{3} s_{n - 5,k} + 20 k^{3} s_{n - 5,k - 2} + 20 k^{3} s_{n - 5,k + 1} + 15 k^{2} s_{n - 5,k} + 15 k^{2} s_{n - 5,k - 3} - 30 k^{2} s_{n - 5,k - 2} + 15 k^{2} s_{n - 5,k - 1} + 15 k^{2} s_{n - 5,k + 1} + 6 k s_{n - 5,k} + 6 k s_{n - 5,k - 4} - 30 k s_{n - 5,k - 3} + 30 k s_{n - 5,k - 2} + 6 k s_{n - 5,k + 1} + s_{n - 5,k} + s_{n - 5,k - 5} - 9 s_{n - 5,k - 4} + 20 s_{n - 5,k - 3} - 10 s_{n - 5,k - 2} + s_{n - 5,k - 1} + s_{n - 5,k + 1}\\
s_{n + 1,k + 1} = k^{7} s_{n - 6,k + 1} + 7 k^{6} s_{n - 6,k} + 7 k^{6} s_{n - 6,k + 1} + 21 k^{5} s_{n - 6,k} + 21 k^{5} s_{n - 6,k - 1} + 21 k^{5} s_{n - 6,k + 1} + 35 k^{4} s_{n - 6,k} + 35 k^{4} s_{n - 6,k - 2} + 35 k^{4} s_{n - 6,k + 1} + 35 k^{3} s_{n - 6,k} + 35 k^{3} s_{n - 6,k - 3} - 70 k^{3} s_{n - 6,k - 2} + 35 k^{3} s_{n - 6,k - 1} + 35 k^{3} s_{n - 6,k + 1} + 21 k^{2} s_{n - 6,k} + 21 k^{2} s_{n - 6,k - 4} - 105 k^{2} s_{n - 6,k - 3} + 105 k^{2} s_{n - 6,k - 2} + 21 k^{2} s_{n - 6,k + 1} + 7 k s_{n - 6,k} + 7 k s_{n - 6,k - 5} - 63 k s_{n - 6,k - 4} + 140 k s_{n - 6,k - 3} - 70 k s_{n - 6,k - 2} + 7 k s_{n - 6,k - 1} + 7 k s_{n - 6,k + 1} + s_{n - 6,k} + s_{n - 6,k - 6} - 14 s_{n - 6,k - 5} + 56 s_{n - 6,k - 4} - 70 s_{n - 6,k - 3} + 21 s_{n - 6,k - 2} + s_{n - 6,k + 1}\\
s_{n + 1,k + 1} = k^{8} s_{n - 7,k + 1} + 8 k^{7} s_{n - 7,k} + 8 k^{7} s_{n - 7,k + 1} + 28 k^{6} s_{n - 7,k} + 28 k^{6} s_{n - 7,k - 1} + 28 k^{6} s_{n - 7,k + 1} + 56 k^{5} s_{n - 7,k} + 56 k^{5} s_{n - 7,k - 2} + 56 k^{5} s_{n - 7,k + 1} + 70 k^{4} s_{n - 7,k} + 70 k^{4} s_{n - 7,k - 3} - 140 k^{4} s_{n - 7,k - 2} + 70 k^{4} s_{n - 7,k - 1} + 70 k^{4} s_{n - 7,k + 1} + 56 k^{3} s_{n - 7,k} + 56 k^{3} s_{n - 7,k - 4} - 280 k^{3} s_{n - 7,k - 3} + 280 k^{3} s_{n - 7,k - 2} + 56 k^{3} s_{n - 7,k + 1} + 28 k^{2} s_{n - 7,k} + 28 k^{2} s_{n - 7,k - 5} - 252 k^{2} s_{n - 7,k - 4} + 560 k^{2} s_{n - 7,k - 3} - 280 k^{2} s_{n - 7,k - 2} + 28 k^{2} s_{n - 7,k - 1} + 28 k^{2} s_{n - 7,k + 1} + 8 k s_{n - 7,k} + 8 k s_{n - 7,k - 6} - 112 k s_{n - 7,k - 5} + 448 k s_{n - 7,k - 4} - 560 k s_{n - 7,k - 3} + 168 k s_{n - 7,k - 2} + 8 k s_{n - 7,k + 1} + s_{n - 7,k} + s_{n - 7,k - 7} - 20 s_{n - 7,k - 6} + 126 s_{n - 7,k - 5} - 294 s_{n - 7,k - 4} + 231 s_{n - 7,k - 3} - 42 s_{n - 7,k - 2} + s_{n - 7,k - 1} + s_{n - 7,k + 1}\\
s_{n + 1,k + 1} = k^{9} s_{n - 8,k + 1} + 9 k^{8} s_{n - 8,k} + 9 k^{8} s_{n - 8,k + 1} + 36 k^{7} s_{n - 8,k} + 36 k^{7} s_{n - 8,k - 1} + 36 k^{7} s_{n - 8,k + 1} + 84 k^{6} s_{n - 8,k} + 84 k^{6} s_{n - 8,k - 2} + 84 k^{6} s_{n - 8,k + 1} + 126 k^{5} s_{n - 8,k} + 126 k^{5} s_{n - 8,k - 3} - 252 k^{5} s_{n - 8,k - 2} + 126 k^{5} s_{n - 8,k - 1} + 126 k^{5} s_{n - 8,k + 1} + 126 k^{4} s_{n - 8,k} + 126 k^{4} s_{n - 8,k - 4} - 630 k^{4} s_{n - 8,k - 3} + 630 k^{4} s_{n - 8,k - 2} + 126 k^{4} s_{n - 8,k + 1} + 84 k^{3} s_{n - 8,k} + 84 k^{3} s_{n - 8,k - 5} - 756 k^{3} s_{n - 8,k - 4} + 1680 k^{3} s_{n - 8,k - 3} - 840 k^{3} s_{n - 8,k - 2} + 84 k^{3} s_{n - 8,k - 1} + 84 k^{3} s_{n - 8,k + 1} + 36 k^{2} s_{n - 8,k} + 36 k^{2} s_{n - 8,k - 6} - 504 k^{2} s_{n - 8,k - 5} + 2016 k^{2} s_{n - 8,k - 4} - 2520 k^{2} s_{n - 8,k - 3} + 756 k^{2} s_{n - 8,k - 2} + 36 k^{2} s_{n - 8,k + 1} + 9 k s_{n - 8,k} + 9 k s_{n - 8,k - 7} - 180 k s_{n - 8,k - 6} + 1134 k s_{n - 8,k - 5} - 2646 k s_{n - 8,k - 4} + 2079 k s_{n - 8,k - 3} - 378 k s_{n - 8,k - 2} + 9 k s_{n - 8,k - 1} + 9 k s_{n - 8,k + 1} + s_{n - 8,k} + s_{n - 8,k - 8} - 27 s_{n - 8,k - 7} + 246 s_{n - 8,k - 6} - 924 s_{n - 8,k - 5} + 1407 s_{n - 8,k - 4} - 735 s_{n - 8,k - 3} + 85 s_{n - 8,k - 2} + s_{n - 8,k + 1}\\\end{array}

In [36]:

    

s = oeis.oeis_search(seq=[1,1,4,3,1,27,19,6,1,256,175,55,10,1])


    

    




*

In [37]:

    

s()


    

    
Out[37]:




Results for query: https://oeis.org/search?q=1%2C+1%2C+4%2C+3%2C+1%2C+27%2C+19%2C+6%2C+1%2C+256%2C+175%2C+55%2C+10%2C+1&start=0&fmt=json
A039621: Triangle of Lehmer-Comtet numbers of 2nd kind.

by Len Smiley
Keywords: tabl,sign
Data:
$$
\begin{array}{c|ccccccccccc}
n, k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
0 & 1 &  &  &  &  &  &  &  &  & \\1 & -1 & 1 &  &  &  &  &  &  &  & \\2 & 4 & -3 & 1 &  &  &  &  &  &  & \\3 & -27 & 19 & -6 & 1 &  &  &  &  &  & \\4 & 256 & -175 & 55 & -10 & 1 &  &  &  &  & \\5 & -3125 & 2101 & -660 & 125 & -15 & 1 &  &  &  & \\6 & 46656 & -31031 & 9751 & -1890 & 245 & -21 & 1 &  &  & \\7 & -823543 & 543607 & -170898 & 33621 & -4550 & 434 & -28 & 1 &  & \\8 & 16777216 & -11012415 & 3463615 & -688506 & 95781 & -9702 & 714 & -36 & 1 & \\9 & -387420489
\end{array}
$$
Comments:

Also the Bell transform of (-n)^n adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 16 2016

Formulae:

(k-1)!a(n, k) = Sum_{i=0..k-1}((-1)^(n-k-i)binomial(k-1, i)*(n-i-1)^(n-1)).

Cross references:

Cf. A008296 (matrix inverse). Also A045531 (for column |a(n, 2)|). A185164.

Links:

D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

In [40]:

    

s = oeis.oeis_search(id=264428)


    

    




*

In [41]:

    

s()


    

    
Out[41]:




Results for query: https://oeis.org/search?q=id%3AA264428&start=0&fmt=json
A264428: Triangle read by rows, Bell transform of Bell numbers.

by Peter Luschny, Nov 13 2015
Keywords: nonn,tabl
Data:
$$
\begin{array}{c|ccccccccccc}
n, k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
0 & 1 &  &  &  &  &  &  &  &  & \\1 & 0 & 1 &  &  &  &  &  &  &  & \\2 & 0 & 1 & 1 &  &  &  &  &  &  & \\3 & 0 & 2 & 3 & 1 &  &  &  &  &  & \\4 & 0 & 5 & 11 & 6 & 1 &  &  &  &  & \\5 & 0 & 15 & 45 & 35 & 10 & 1 &  &  &  & \\6 & 0 & 52 & 205 & 210 & 85 & 15 & 1 &  &  & \\7 & 0 & 203 & 1029 & 1330 & 700 & 175 & 21 & 1 &  & \\8 & 0 & 877 & 5635 & 8946 & 5845 & 1890 & 322 & 28 & 1 & \\9 & 0 & 4140 & 33387 & 63917 & 50358 & 20055 & 4410 & 546 & 36 & 1
\end{array}
$$
Comments:

Consider the sequence S0 -> T0 -> S1 -> T1 -> S2 -> T2 -> ... Here Sn -> Tn indicates the Bell transform mapping a sequence Sn to a triangle Tn as defined in the link and Tn -> S{n+1} the operator associating a triangle with the sequence of its row sums. If
S0 = A000012 = <1,1,1,...> then
T0 = A048993 # Stirling subset numbers,
S1 = A000110 # Bell numbers,
T1 = A264428 # Bell transform of Bell numbers,
S2 = A187761 # second-order Bell numbers,
T2 = A264430 # Bell transform of second-order Bell numbers,
S3 = A264432 # third-order Bell numbers.
This construction is closely related to permutations trees and A179455. Sn is A179455_col(n+1) prepended by A179455_diag(k) = k! for k <= n. In other words, Sn 'converges' to n! for n -> inf.
Given a sequence (s(n))n>=0 with s(0) = 0 and with e.g.f. B(x) = Sum_{n >= 1} s(n)*x^n/n!, then the Bell matrix associated with s(n) equals the exponential Riordan array [1, B(x)] belonging to the Lagrange subgroup of the exponential Riordan group. Omitting the first row and column from the Bell matrix produces the exponential Riordan array [d/dx(B(x)), B(x)] belonging to the Derivative subgroup of the exponential Riordan group. - Peter Bala, Jun 07 2016

Formulae:

From Peter Bala, Jun 07 2016: 

E.g.f. exp(tB(x)), where B(x) = Integral_{u = 0..x} exp(exp(u) - 1) du =  x + x^2/2! + 2x^3/3! + 5x^4/4! + 15x^5/5! + 52*x^6/6! + ....
Row polynomial recurrence: R(n+1,t) = tSum_{k = 0 ..n} binomial(n,k)Bell(k)* R(n-k,t) with R(0,t) = 1. 



Cross references:

Cf. A000012, A000110, A000217, A000914, A027801, A048993, A051836, A179455, A187761 (row sums), A264430, A264432, A265312.

Links:

G. C. Greubel, <a href="/A264428/b264428.txt">Table of n, a(n) for n = 0..1325</a>
Peter Luschny, The Bell transform
Peter Luschny, Permutation Trees