A304655 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k^3*(n-k)^2).

0, 0, 8, 81, 2480, 175000, 23825904, 5563712448, 2051674085376, 1124193889529856, 873600549068759040, 927968580453961728000, 1307864687259363065856000, 2386263863328126193631232000, 5521179117888960788194394112000, 15917227342113559040727019683840000

Offset: 0

Author

Vaclav Kotesovec, May 16 2018

Keywords

• nonn

Crossrefs

Cf. A052517, A302827, A304589, A304654.

Programs

Mathematica

Table[(n!)^3 * Sum[1/(k^3*(n-k)^2), {k, 1, n-1}], {n, 0, 20}]

Formula

Recurrence: n*(12*n^4 - 108*n^3 + 354*n^2 - 501*n + 260)*a(n) = 2*(n-1)*(24*n^7 - 306*n^6 + 1620*n^5 - 4599*n^4 + 7516*n^3 - 7015*n^2 + 3444*n - 696)*a(n-1) - 6*(n-2)^4*(12*n^7 - 162*n^6 + 906*n^5 - 2700*n^4 + 4583*n^3 - 4378*n^2 + 2163*n - 436)*a(n-2) + 2*(n-3)^4*(n-2)^3*(24*n^7 - 342*n^6 + 2004*n^5 - 6201*n^4 + 10816*n^3 - 10497*n^2 + 5208*n - 1048)*a(n-3) - (n-4)^5*(n-3)^5*(n-2)^3*(12*n^4 - 60*n^3 + 102*n^2 - 69*n + 17)*a(n-4).

a(n)/(n!)^3 ~ Zeta(3)/n^2.