A309386 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} (-k)^(n-j)*Stirling2(n,j).

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 1, 1, 1, 1, -3, 1, 9, 2, 1, 1, 1, -4, 5, 19, -23, -9, 1, 1, 1, -5, 11, 25, -128, -25, 9, 1, 1, 1, -6, 19, 21, -343, 379, 583, 50, 1, 1, 1, -7, 29, 1, -674, 2133, 1549, -3087, -267, 1

Offset: 0

Author

Seiichi Manyama, Jul 27 2019

Keywords

• sign
• tabl

Examples

			Square array begins:
   1,  1,   1,    1,    1,    1,     1, ...
   1,  1,   1,    1,    1,    1,     1, ...
   1,  0,  -1,   -2,   -3,   -4,    -5, ...
   1, -1,  -1,    1,    5,   11,    19, ...
   1,  1,   9,   19,   25,   21,     1, ...
   1,  2, -23, -128, -343, -674, -1103, ...
   1, -9, -25,  379, 2133, 6551, 15211, ...
		

Links

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

Crossrefs

Columns k=0..6 give A000012, (-1)^n * A000587(n), A009235, A317996, A318179, A318180, A318181.

Rows n=0+1, 2 give A000012, A024000.

Main diagonal gives A318183.

Cf. A241578, A292861.

Programs

Mathematica

T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * StirlingS2[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 07 2021 *)

Formula

E.g.f. of column k: exp((1 - exp(-k*x))/k) for k > 0.

A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} (-k)^(n-1-j) * binomial(n-1,j) * A(j,k) for n > 0.