A370226 -id:A370226 - 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.

A370225 a(n) = n!^2 * [x^n] polylog(2,x)^3.

Original entry on oeis.org

0, 0, 0, 36, 432, 7500, 191700, 6899592, 334858944, 21143500992, 1687177771200, 166164785309952, 19807714583407872, 2811642170175940608, 468723024212650002432, 90691236890980830683136, 20158684374653040962764800, 5101828835401918167928012800, 1458612909355515900114154291200

Offset: 0

Vaclav Kotesovec, Feb 12 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Polylogarithm.
Wikipedia, Polylogarithm.

Cf. A302827, A370226.

CoefficientList[Series[PolyLog[2, x]^3, {x, 0, 20}], x] * Range[0, 20]!^2
Table[n!^2 * Sum[Sum[1/(k*(j-k))^2, {k, 1, j-1}] * 1/(n-j)^2, {j, 1, n-1}], {n, 0, 20}]

Recurrence: (n-1) * n^2 * (24*n^9 - 648*n^8 + 7928*n^7 - 58928*n^6 + 297426*n^5 - 1054693*n^4 + 2579632*n^3 - 4091937*n^2 + 3732024*n - 1463688) * a(n) = 4*(n-1)^2 * (36*n^12 - 1146*n^11 + 16806*n^10 - 151862*n^9 + 950049*n^8 - 4345179*n^7 + 14812504*n^6 - 37460061*n^5 + 68666655*n^4 - 87462969*n^3 + 72065619*n^2 - 33585192*n + 6382800) * a(n-1) - (n-2)^2 * (360*n^14 - 13560*n^13 + 237600*n^12 - 2585176*n^11 + 19644502*n^10 - 110806171*n^9 + 478897261*n^8 - 1605269146*n^7 + 4161296764*n^6 - 8214107379*n^5 + 12005416617*n^4 - 12428493288*n^3 + 8478515088*n^2 - 3337578000*n + 550726128) * a(n-2) + 8*(n-3)^4 * (60*n^14 - 2370*n^13 + 43430*n^12 - 492632*n^11 + 3892949*n^10 - 22819863*n^9 + 102674094*n^8 - 359603394*n^7 + 978009142*n^6 - 2031308188*n^5 + 3125785870*n^4 - 3399739356*n^3 + 2424585966*n^2 - 989615976*n + 166980888) * a(n-3) - (n-4)^4 * (n-3)^2 * (360*n^14 - 14880*n^13 + 284280*n^12 - 3348872*n^11 + 27395534*n^10 - 165978041*n^9 + 772288075*n^8 - 2803517230*n^7 + 7923458764*n^6 - 17127191793*n^5 + 27415542063*n^4 - 30937263696*n^3 + 22788329436*n^2 - 9544987584*n + 1636611264) * a(n-4) + 4*(n-5)^5 * (n-4)^2 * (n-3)^2 * (36*n^13 - 1374*n^12 + 24000*n^11 - 256474*n^10 + 1894307*n^9 - 10337820*n^8 + 43164880*n^7 - 139027139*n^6 + 340176690*n^5 - 610956909*n^4 + 764185680*n^3 - 614613168*n^2 + 276176448*n - 49654512) * a(n-5) - (n-6)^6 * (n-5)^5 * (n-4)^2 * (n-3)^2 * (24*n^9 - 432*n^8 + 3608*n^7 - 19560*n^6 + 77082*n^5 - 216339*n^4 + 399768*n^3 - 441651*n^2 + 252364*n - 52860) * a(n-6).

a(n)/(n!)^2 ~ Pi^4 / (12*n^2) * (1 + 24*log(n)/(Pi^2*n)).

cp's OEIS Frontend

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

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