A382740 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).

1, 1, 1, 1, 7, 1, 1, 19, 19, 1, 1, 43, 127, 43, 1, 1, 91, 559, 559, 91, 1, 1, 187, 2071, 4327, 2071, 187, 1, 1, 379, 7039, 25831, 25831, 7039, 379, 1, 1, 763, 22807, 133783, 233551, 133783, 22807, 763, 1, 1, 1531, 71839, 636679, 1748791, 1748791, 636679, 71839, 1531, 1

Offset

1,5

Links

Table of n, a(n) for n=1..55.

Formula

E.g.f.: (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).

A(n,k) = A(k,n).

A(n,k) = (1/2) * A382734(n,k).

Example

Square array begins:
  1,   1,    1,      1,       1,        1, ...
  1,   7,   19,     43,      91,      187, ...
  1,  19,  127,    559,    2071,     7039, ...
  1,  43,  559,   4327,   25831,   133783, ...
  1,  91, 2071,  25831,  233551,  1748791, ...
  1, 187, 7039, 133783, 1748791, 18207367, ...

Prog

(PARI) a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*stirling(n, j, 2)*stirling(k, j, 2))/2;

Crossrefs

Cf. A272644, A382741, A382742.

Cf. A382734.

Sequence in context: A391048 A176284 A154233 * A174033 A119727 A157272

Adjacent sequences: A382737 A382738 A382739 * A382741 A382742 A382743

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 04 2025

Status

approved