A382742 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/4) * (1 / (exp(x) + exp(y) - exp(x+y))^4 - 1).

1, 1, 1, 1, 11, 1, 1, 31, 31, 1, 1, 71, 271, 71, 1, 1, 151, 1291, 1291, 151, 1, 1, 311, 4951, 12011, 4951, 311, 1, 1, 631, 17131, 78451, 78451, 17131, 631, 1, 1, 1271, 56071, 426971, 820351, 426971, 56071, 1271, 1, 1, 2551, 177691, 2093491, 6709651, 6709651, 2093491, 177691, 2551, 1

Offset

1,5

Links

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

Formula

E.g.f.: (1/4) * (1 / (exp(x) + exp(y) - exp(x+y))^4 - 1).

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

A(n,k) = (1/4) * A382736(n,k).

Example

Square array begins:
  1,   1,     1,      1,       1,        1, ...
  1,  11,    31,     71,     151,      311, ...
  1,  31,   271,   1291,    4951,    17131, ...
  1,  71,  1291,  12011,   78451,   426971, ...
  1, 151,  4951,  78451,  820351,  6709651, ...
  1, 311, 17131, 426971, 6709651, 79008011, ...

Prog

(PARI) a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+3, 3)*stirling(n, j, 2)*stirling(k, j, 2))/4;

Crossrefs

Cf. A272644, A382740, A382741.

Cf. A382736.

Sequence in context: A060270 A202678 A202971 * A202675 A176198 A202870

Adjacent sequences: A382739 A382740 A382741 * A382743 A382744 A382745

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 04 2025

Status

approved