A373772 Expansion of e.g.f. exp(x^3 / (6 * (1 - x))) / (1 - x).

1, 1, 2, 7, 32, 180, 1210, 9450, 84000, 836920, 9234400, 111742400, 1471023400, 20925905000, 319830310800, 5226116295400, 90906373958400, 1676967192700800, 32697692264036800, 671856896755844800, 14509136903381120000, 328520930667097168000

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..446

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k,n-3*k)/(6^k * k!).

D-finite with recurrence: 2*(n + 1)*(n + 2)*(n + 3)*a(n) - 3*(n + 2)*(n + 3)*a(n+1) + 6*(n + 3)^2*a(n + 2) - 6*(7 + 2*n)*a(n + 3) + 6*a(n + 4). - Robert Israel, May 06 2026

Maple

f:= gfun:-rectoproc({{2*a(n)*(n + 1)*(n + 2)*(n + 3) - 3*a(n + 1)*(n + 2)*(n + 3) + 6*(n + 3)^2*a(n + 2) - 6*(7 + 2*n)*a(n + 3) + 6*a(n + 4), a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 7}, a(n), remember):
map(f, [$0..30]); # Robert Israel, May 06 2026

Prog

(PARI) a(n) = n!*sum(k=0, n\3, binomial(n-2*k, n-3*k)/(6^k*k!));

Crossrefs

Cf. A130906, A361597, A373773.

Cf. A000930, A361533.

Sequence in context: A351813 A301465 A097900 * A375395 A198891 A000153

Adjacent sequences: A373769 A373770 A373771 * A373773 A373774 A373775

Keyword

nonn

Author

Seiichi Manyama, Jun 18 2024

Status

approved