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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl

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

Examples

			From _Gheorghe Coserea_, Jan 30 2017: (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)
		

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.

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.

Programs

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 *)

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

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