A301992 a(n) = 8*(n-2)*(2*n-5)*a(n-1) + ((n-2)/9)*Product_{k=0..n-2} (2*k-3)^2 with a(1) = 0.

0, 0, 1, 50, 6027, 1350948, 486396405, 256822659990, 186967652864895, 179489092842045000, 219694686618136235625, 333935935534086791456250, 617113613582364168765061875, 1362586861058382580086938587500, 3542725840051847141662287901708125

Offset

1,4

Links

Table of n, a(n) for n=1..15.

Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 10, (3.43) - (3.47).

Formula

a(n) = (f2(n-1)/8)*Product_{k=0..n-2} (2*k-1)^2 where f2(n) corresponds to the y values such that Sum_{k>=0} 1/(binomial(2*k,k)*(2*k+(2*n-1))) = x*Pi*sqrt(3) - y. (See examples for connection with a(n) in terms of material at Links section).

Example

Examples ((3.43) - (3.47)) at page 10 in Links section as follows, respectively.
For n=1, f2(1) = 0, so a(2) = 0.
For n=2, f2(2) = 8, so a(3) = 1.
For n=3, f2(3) = 400/9, so a(4) = 50.
For n=4, f2(4) = 16072/75, so a(5) = 6027.
For n=5, f2(5) = 3602528/3675, so a(6) = 1350948.

Mathematica

RecurrenceTable[{b[n + 1] == 8*(n - 1)*(2*n - 3)*b[n] + (n - 1)/9 * Product[(2*k - 3)^2, {k, 0, n - 1}], b[1] == 0}, b, {n, 1, 20}] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 15; Flatten[{0, Table[CoefficientList[Expand[FunctionExpand[Table[ FullSimplify[-Sum[1/(Binomial[2*j, j]*(2*j + (2*m - 1))), {j, 0, Infinity}]]*Product[(2*k - 1)^2, {k, 0, m - 1}]/8, {m, 0, nmax}]]], Pi][[n, 1]], {n, 2, nmax}]}] (* Vaclav Kotesovec, Apr 12 2018 *)

Prog

(PARI) a=vector(20); a[1]=0; for(n=2, #a, a[n]=8*(n-2)*(2*n-5)*a[n-1] + (n-2)*prod(k=0, n-2, (2*k-3)^2)/9); a \\ Altug Alkan, Mar 30 2018

Crossrefs

Cf. A000984, A301990.

Sequence in context: A320099 A115436 A184555 * A229753 A276102 A152258

Adjacent sequences: A301989 A301990 A301991 * A301993 A301994 A301995

Keyword

nonn

Author

Detlef Meya, Mar 30 2018

Extensions

More terms from Vaclav Kotesovec, Mar 30 2018

Status

approved