A060080 Scaled sums of squares.

1, 5, 56, 1080, 31680, 1310400, 72576000, 5181926400, 463325184000, 50697529344000, 6663103856640000, 1035678099456000000, 187913434365296640000, 39357424864287129600000, 9424067526124476825600000, 2558178330075402338304000000, 781407199004850168791040000000

Offset: 1

Author

Wolfdieter Lang, Mar 16 2001

Keywords

• nonn

Links

• Harry J. Smith, Table of n, a(n) for n = 1..201 [offset shifted by _Georg Fischer_, May 11 2021]

Crossrefs

See Comments section at A060074.

Cf. A000330, A001044.

Programs

Magma

[Factorial(n+1) * Factorial(n-1) * (2*n+1)/6: n in [1..20]]; // Vincenzo Librandi, Jul 05 2018

Mathematica

Table[(n + 1)! (n - 1)! (2 n + 1) / 3!, {n, 1, 30}] (* Vincenzo Librandi Jul 04 2018 *)

PARI

a(n) = { (n + 1)!*(n - 1)!*(2*n + 1)/6 } \\ Harry J. Smith, Jul 01 2009

Formula

a(n) = (Sum_{k=1..n} k^2)*((n-1)!)^2 = (n+1)!*(n-1)!*(2*n+1)/3!.

a(n) = A000330(n) * A001044(n-1).

From Amiram Eldar, May 03 2025: (Start)

Sum_{n>=1} 1/a(n) = 3 * BesselI(1, 2) (Pi * StruveL(0, 2) + 2) - 3 * BesselI(0, 2) * (Pi * StruveL(1, 2) + 2).

Sum_{n>=1} (-1)^(n+1)/a(n) = 3 * BesselJ(1, 2) * (Pi * StruveH(0, 2) - 2) - 3 * BesselJ(0, 2) * (Pi * StruveH(1, 2) - 2). (End)

Extensions

Offset changed from 0 to 1 by Georg Fischer, May 09 2021