A106800 - OEIS

A106800

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

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, 140, 350, 301, 63, 1, 0, 1, 28, 266, 1050, 1701, 966, 127, 1, 0, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 0, 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

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

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.

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

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, table 2.14.1 at page 24.

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv preprint arXiv:1402.2960 [math.CO], 2014.

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].

J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.

Eric Weisstein's World of Mathematics, Bell Polynomial.

FORMULA

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

Also, P_n(t) * exp(t) = (t * d/dt)^n exp(t). - Michael Somos, Aug 16 2017

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

EXAMPLE

From Gheorghe Coserea, Jan 30: (Start)

Triangle starts:

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

[0] 1;

[1] 1, 0;

[2] 1, 1, 0;

[3] 1, 3, 1, 0;

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

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

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

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

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

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

...

(End)

MAPLE

seq(seq(Stirling2(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021

MATHEMATICA

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

PROG

(PARI)

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

{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 */

(Sage) flatten([[stirling_number2(n, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 11 2021

CROSSREFS

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

The Stirling1 counterpart is A054654.

Row sum: A000110.

Column 0: A000012.

Column 1: A000217.

Main Diagonal: A000007.

1st minor diagonal: A000012.

2nd minor diagonal: A000225.

3rd minor diagonal: A000392.

Cf. A008278, A340556.