A091032 Second column (k=3) of array A090438 ((4,2)-Stirling2) divided by 8.

1, 60, 5040, 604800, 99792000, 21794572800, 6102480384000, 2134124568576000, 912338253066240000, 468333636574003200000, 284372184127734743040000, 201645730563302817792000000, 165147853331345007771648000000

Offset: 2

Author

Wolfdieter Lang, Jan 23 2004

Keywords

• nonn
• easy

Links

• John Campbell, Applications of Gosper’s nonlocal derangement identity, Maple Transactions, 5, 1, Article 16724, Feb. 2025.

Crossrefs

Cf. A002674 (first column of A090438), A091033 (third column), A090438.

Cf. A001620, A049470, A073742, A073743, A099281, A099282, A099284.

Programs

Mathematica

a[n_] := (n - 1)*(2*n)!/4!; Array[a, 13, 2] (* Amiram Eldar, Nov 03 2022 *)

PARI

a(n) = (n-1)*(2*n)!/4!; \\ Amiram Eldar, Nov 03 2022

Formula

a(n) = A090438(n, 3)/8 = (n-1)*(2*n)!/4!

E.g.f.: (-3*hypergeom([1/2, 1], [], 4*x) + hypergeom([1, 3/2], [], 4*x) + 2)/(8*3!) (cf. A090438).

From Amiram Eldar, Nov 03 2022: (Start)

Sum_{n>=2} 1/a(n) = 60 - 24*Gamma - 24*cosh(1) + 24*CoshIntegral(1) - 24*sinh(1).

Sum_{n>=2} (-1)^n/a(n) = -12 + 24*gamma - 24*cos(1) - 24*CosIntegral(1) + 24*SinIntegral(1). (End)

a(n+1) = Sum_{j = 1..n} (-1)^(n+j) * j^(2*n+2) * binomial(2*n, n-j) (Campbell, Eq. 17). - Peter Bala, Mar 30 2025