A279636 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.

1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 5, 10, 15, 1, 1, 9, 22, 41, 52, 1, 1, 17, 52, 125, 196, 203, 1, 1, 33, 130, 413, 836, 1057, 877, 1, 1, 65, 340, 1445, 3916, 6277, 6322, 4140, 1, 1, 129, 922, 5261, 19676, 41077, 52396, 41393, 21147, 1, 1, 257, 2572, 19685, 104116, 288517, 481384, 479593, 293608, 115975

Offset: 0

Author

Alois P. Heinz, Dec 16 2016

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
:   1,    1,    1,     1,      1,       1,        1, ...
:   1,    1,    1,     1,      1,       1,        1, ...
:   2,    3,    5,     9,     17,      33,       65, ...
:   5,   10,   22,    52,    130,     340,      922, ...
:  15,   41,  125,   413,   1445,    5261,    19685, ...
:  52,  196,  836,  3916,  19676,  104116,   572036, ...
: 203, 1057, 6277, 41077, 288517, 2133397, 16379797, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• Wikipedia, Kronecker delta

Crossrefs

Columns k=0-10 give: A000110, A000248, A033462, A279358, A279637, A279638, A279639, A279640, A279641, A279642, A279643.

Rows n=0+1,2 give: A000012, A000051.

Main diagonal gives A279644.

Cf. A145460.

Programs

Maple

egf:= k-> exp(exp(x)*add(Stirling2(k, j)*x^j, j=0..k)-`if`(k=0, 1, 0)):
A:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^k*A(n-j, k), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[Binomial[n-1, j-1]*j^k*A[n-j, k], {j, 1, n}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

Formula

E.g.f. of column k: exp(exp(x)*(Sum_{j=0..k} Stirling2(n,j)*x^j) - delta_{0,k}).