A373771 Expansion of e.g.f. exp(x^2 / (2 * (1 - x)^3)) / (1 - x).

1, 1, 3, 18, 147, 1425, 15855, 200130, 2838465, 44767485, 777046095, 14705245170, 301014595035, 6621102973485, 155640761791515, 3891902825660850, 103115436832433025, 2884715829245475225, 84950805438277854075, 2626194012669689512050

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..426

Formula

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

D-finite with recurrence: 2*(n + 1)^2*(n + 2)*(n + 3)*a(n) - (15 + 8*n)*(n + 2)*(n + 3)*a(n+1) + 4*(7 + 3*n)*(n + 3)*a(n+2) - 2*(13 + 4*n)*a(n + 3) + 2*a(n + 4) = 0. - Robert Israel, May 06 2026

Maple

f:= gfun:-rectoproc({2*(n + 1)^2*(n + 2)*(n + 3)*a(n) - (15 + 8*n)*(n + 2)*(n + 3)*a(n+1) + 4*(7 + 3*n)*(n + 3)*a(n+2) - 2*(13 + 4*n)*a(n + 3) + 2*a(n + 4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 18}, a(n), remember):
map(f, [$0..25]); # Robert Israel, May 06 2026

Prog

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

Crossrefs

Cf. A130905, A361596, A373770.

Cf. A361597, A361599.

Cf. A335345.

Sequence in context: A118331 A371825 A240921 * A379249 A221961 A385500

Adjacent sequences: A373768 A373769 A373770 * A373772 A373773 A373774

Keyword

nonn

Author

Seiichi Manyama, Jun 18 2024

Status

approved