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Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
5

%I #50 Sep 11 2021 22:26:21

%S 1,1,0,1,1,0,1,3,1,0,1,6,7,1,0,1,10,25,15,1,0,1,15,65,90,31,1,0,1,21,

%T 140,350,301,63,1,0,1,28,266,1050,1701,966,127,1,0,1,36,462,2646,6951,

%U 7770,3025,255,1,0,1,45,750,5880,22827,42525,34105,9330,511,1,0

%N Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.

%C Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, May 19 2005

%D 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.

%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.

%H G. C. Greubel, <a href="/A106800/b106800.txt">Rows n = 0..50 of the triangle, flattened</a>

%H A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, O. Mallet, <a href="http://arxiv.org/abs/1402.2960">Bell polynomials in combinatorial Hopf algebras</a>, arXiv preprint arXiv:1402.2960 [math.CO], 2014.

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>

%F A(x;t) = exp(t*(exp(x)-1)) = Sum_{n>=0} P_n(t) * x^n/n!, where P_n(t) = Sum_{k=0..n} T(n,k)*t^(n-k). - _Gheorghe Coserea_, Jan 30 2017

%F Also, P_n(t) * exp(t) = (t * d/dt)^n exp(t). - _Michael Somos_, Aug 16 2017

%F T(n, k) = Sum_{j=0..k} E2(k, j)*binomial(n + k - j, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - _Peter Luschny_, Feb 21 2021

%e From _Gheorghe Coserea_, Jan 30: (Start)

%e Triangle starts:

%e n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9]

%e [0] 1;

%e [1] 1, 0;

%e [2] 1, 1, 0;

%e [3] 1, 3, 1, 0;

%e [4] 1, 6, 7, 1, 0;

%e [5] 1, 10, 25, 15, 1, 0;

%e [6] 1, 15, 65, 90, 31, 1, 0;

%e [7] 1, 21, 140, 350, 301, 63, 1, 0;

%e [8] 1, 28, 266, 1050, 1701, 966, 127, 1, 0;

%e [9] 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 0;

%e ...

%e (End)

%p seq(seq(Stirling2(n, n-k), k=0..n), n=0..8); # _Peter Luschny_, Feb 21 2021

%t Table[ StirlingS2[n, m], {n, 0, 10}, {m, n, 0, -1}]//Flatten (* _Robert G. Wilson v_, Jan 30 2017 *)

%o (PARI)

%o N=11; x='x+O('x^N); t='t; concat(apply(p->Vec(p), Vec(serlaplace(exp(t*(exp(x)-1)))))) \\ _Gheorghe Coserea_, Jan 30 2017

%o {T(n, k) = my(A, B); if( n<0 || k>n, 0, A = B = exp(x + x * O(x^n)); for(i=1, n, A = x * A'); polcoeff(A / B, n-k))}; /* _Michael Somos_, Aug 16 2017 */

%o (Sage) flatten([[stirling_number2(n, n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 11 2021

%Y See A008277 and A048993, which are the main entries for this triangle of numbers.

%Y The Stirling1 counterpart is A054654.

%Y Row sum: A000110.

%Y Column 0: A000012.

%Y Column 1: A000217.

%Y Main Diagonal: A000007.

%Y 1st minor diagonal: A000012.

%Y 2nd minor diagonal: A000225.

%Y 3rd minor diagonal: A000392.

%Y Cf. A008278, A340556.

%K nonn,tabl

%O 0,8

%A _N. J. A. Sloane_