A293119 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(-x^i).

1, 1, -1, 1, 0, -1, 1, 0, -2, -1, 1, 0, 0, -6, 1, 1, 0, 0, -6, -12, 19, 1, 0, 0, 0, -24, 0, 151, 1, 0, 0, 0, -24, -120, 240, 1091, 1, 0, 0, 0, 0, -120, -360, 2520, 7841, 1, 0, 0, 0, 0, -120, -720, 0, 21840, 56519, 1, 0, 0, 0, 0, 0, -720, -5040, 20160, 181440, 396271

Offset: 0

Author

Seiichi Manyama, Sep 30 2017

Keywords

• sign
• tabl
• look

Examples

			Square array begins:
    1,   1,    1,    1, ...
   -1,   0,    0,    0, ...
   -1,  -2,    0,    0, ...
   -1,  -6,   -6,    0, ...
    1, -12,  -24,  -24, ...
   19,   0, -120, -120, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Columns k=0..2 give A293116, A293117, A293118.

Rows n=0..1 give A000012, (-1)*A000007.

Main diagonal gives A000007.

A(n,n-1) gives (-1)*A000142(n).

Cf. A293053.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, -add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 30 2017

Mathematica

A[0, _] = 1;
A[n_, k_] /; 0 <= k <= n := A[n, k] = -Sum[A[n-j, k] Binomial[n-1, j-1] j!, {j, k+1, n}];
A[, ] = 0;
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)

Formula

E.g.f. of column k: exp(x^(k+1)/(x-1)).

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = - Sum_{i=k..n-1} (i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.