A369738 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(1 - (1+x)^k).

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 2, -1, 0, 1, -4, 3, 4, 1, 0, 1, -5, 4, 21, -20, -1, 0, 1, -6, 5, 56, -63, 8, 1, 0, 1, -7, 6, 115, -104, -423, 184, -1, 0, 1, -8, 7, 204, -95, -2464, 1899, -464, 1, 0, 1, -9, 8, 329, 36, -8245, 1696, 15201, -1648, -1, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

T(0,k) = 1; T(n,k) = -k * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1) * T(n-j,k)/(n-j)!.

T(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * A000587(j).

Example

Square array T(n,k) begins:
  1,  1,   1,    1,     1,      1,       1, ...
  0, -1,  -2,   -3,    -4,     -5,      -6, ...
  0,  1,   2,    3,     4,      5,       6, ...
  0, -1,   4,   21,    56,    115,     204, ...
  0,  1, -20,  -63,  -104,    -95,      36, ...
  0, -1,   8, -423, -2464,  -8245,  -21096, ...
  0,  1, 184, 1899,  1696, -21275, -124344, ...

Prog

(PARI) a000587(n) = sum(k=0, n, (-1)^k*stirling(n, k, 2));
T(n, k) = sum(j=0, n, k^j*stirling(n, j, 1)*a000587(j));

Crossrefs

Columns k=0..5 give A000007, A033999, (-1)^n * A062267(n), A369751, A369752, A369753.

Main diagonal gives A369754.

Cf. A000587, A294042.

Sequence in context: A256140 A321391 A244003 * A360763 A332670 A118344

Adjacent sequences: A369735 A369736 A369737 * A369739 A369740 A369741

Keyword

sign,tabl

Author

Seiichi Manyama, Jan 30 2024

Status

approved