A304581 -id:A304581 - 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.

A304589 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k*(n-k))^3.

Original entry on oeis.org

0, 0, 8, 54, 1240, 70000, 7941968, 1589632128, 512918521344, 249820864339968, 174720109813751808, 168721560082538496000, 217977447876560510976000, 367117517435096337481728000, 788739873984137255456342016000, 2122296978948474538763602624512000

Offset: 0

Vaclav Kotesovec, May 15 2018

nonn

In general, for m > 1, Sum_{k=1..n-1} 1/(k*(n-k))^m is asymptotic to 2*Zeta(m)/n^m.

Seiichi Manyama, Table of n, a(n) for n = 0..150
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A052517, A302827, A002547, A002548, A304581, A304582, A304655.

List([0..20],n->Factorial(n)^3*Sum([1..n-1],k->1/(k*(n-k))^3)); # Muniru A Asiru, May 16 2018

seq(factorial(n)^3*add(1/(k*(n-k))^3,k=1..n-1),n=0..20); # Muniru A Asiru, May 16 2018

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

Recurrence: n^2*(12*n^4 - 108*n^3 + 354*n^2 - 501*n + 260)*a(n) = 2*(n-1)^2*(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)^5*(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)^5*(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)^6*(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 ~ 2*Zeta(3)/n^3.

A304582 Denominator of Sum_{k=1..n-1} 1/(k*(n-k))^2.

Original entry on oeis.org

1, 1, 1, 2, 144, 72, 64800, 1800, 2822400, 3175200, 317520000, 635040, 11064936960, 15367968, 2545242860160, 14609174580000, 8310997094400, 259718659200, 24319016372851200, 75058692508800, 10838475198270720000, 2389883781218693760, 374701571140216320

Offset: 0

Vaclav Kotesovec, May 15 2018

			0, 0, 1, 1/2, 41/144, 13/72, 8009/64800, 161/1800, 190513/2822400, ...

Eric Weisstein's World of Mathematics, Dilogarithm.
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A002548, A304581, A302827.

CoefficientList[Series[PolyLog[2, x]^2, {x, 0, 25}], x]//Denominator
Table[Sum[1/(k*(n - k))^2, {k, 1, n - 1}], {n, 0, 25}]//Denominator

cp's OEIS Frontend

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

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A304589 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k*(n-k))^3.

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A304582 Denominator of Sum_{k=1..n-1} 1/(k*(n-k))^2.

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