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
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
Programs
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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}).