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

0,8

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.

Links

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

A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, 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, 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.

Sequence in context: A144645 A151510 A151512 * A308484 A227320 A318507

Adjacent sequences: A106797 A106798 A106799 * A106801 A106802 A106803

Keyword

nonn,tabl

Author

N. J. A. Sloane

Status

approved