A304654 -id:A304654 - cp's OEIS frontend

A302827 a(n) = (n!)^2 * Sum_{k=1..n-1} 1/(k*(n-k))^2.

0, 0, 4, 18, 164, 2600, 64072, 2272032, 109735488, 6930012672, 554528623104, 54840436992000, 6568892183808000, 937223951339520000, 157057344897601536000, 30545188599606047539200, 6823697557721234964480000, 1735362552287102663393280000

Offset: 0

Vaclav Kotesovec, May 15 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A052517, A304581, A304582, A304589, A304654, A370225, A370226.

List([0..20],n->Factorial(n)^2*Sum([1..n-1],k->1/(k*(n-k))^2)); # Muniru A Asiru, May 16 2018

seq(factorial(n)^2*add(1/(k*(n-k))^2,k=1..n-1),n=0..20); # Muniru A Asiru, May 16 2018

Table[n!^2*Sum[1/(k*(n-k))^2, {k, 1, n-1}], {n, 0, 20}]
CoefficientList[Series[PolyLog[2, x]^2, {x, 0, 20}], x] * Range[0,20]!^2

Recurrence: n*(2*n - 3)*a(n) = (n-1)*(6*n^3 - 25*n^2 + 33*n - 12)*a(n-1) - (n-2)^3*(6*n^3 - 29*n^2 + 42*n - 15)*a(n-2) + (n-3)^4*(n-2)^3*(2*n - 1)*a(n-3).

a(n) / (n!)^2 ~ Pi^2/(3*n^2) + 4*log(n)/n^3.

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

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 16 2018

nonn

Cf. A052517, A302827, A304589, A304654.

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

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.

cp's OEIS Frontend

A302827 a(n) = (n!)^2 * Sum_{k=1..n-1} 1/(k*(n-k))^2.

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