A293669 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} x^j).
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 1, 3, 13, 25, 1, 1, 1, 3, 13, 49, 81, 1, 1, 1, 3, 13, 73, 261, 331, 1, 1, 1, 3, 13, 73, 381, 1531, 1303, 1, 1, 1, 3, 13, 73, 501, 2611, 9073, 5937, 1, 1, 1, 3, 13, 73, 501, 3331, 19993, 63393, 26785, 1, 1, 1, 3, 13, 73, 501, 4051, 27553, 165873, 465769, 133651, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, ... 1, 3, 3, 3, 3, ... 1, 7, 13, 13, 13, ... 1, 25, 49, 73, 73, ... 1, 81, 261, 381, 501, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Crossrefs
Programs
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Maple
A:= proc(n, k) option remember; `if`(n=0, 1, add( A(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n, k))) end: seq(seq(A(n, 1+d-n), n=0..d), d=0..12); # Alois P. Heinz, Nov 11 2020 -
Mathematica
A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 1] := A[n, k] = (n-1)!*Sum[j*A[n-j, k]/(n-j)!, {j, 1, Min[k, n]}]; A[, ] = 0; Table[A[n, d-n+1], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)
Formula
A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.