A302116 a(n) = 16*(n-1)*((2*n-3)*a(n-1) + (((-1)^n)/9)*Product_{k=0..n-1} (2*k-3)^2) with a(0) = 0.

0, 0, 16, 1504, 361392, 161889216, 116561117520, 123088454370720, 179216801666043120, 344096256861334857600, 842347637388219301894800, 2560736817470194362347292000, 9464483277445284311686053822000, 41795158153162079160478354766472000

Offset

0,3

Links

Table of n, a(n) for n=0..13.

Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 11, (3.53) - (3.57).

Formula

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

Example

Examples ((3.53) - (3.57)) at page 11 in Links section as follows, respectively.
For n=1, f2(1) = 0, so a(1) = 0.
For n=2, f2(2) = 16, so a(2) = 16.
For n=3, f2(3) = -1504/9, so a(3) = 1504.
For n=4, f2(4) = 120464/75, so a(4) = 361392.
For n=5, f2(5) = -53963072/3675, so a(5) = 161889216.

Mathematica

RecurrenceTable[{a[n] == 16*(n-1)*((2*n-3)*a[n-1] + (-1)^n / 9 * Product[(2*k-3)^2, {k, 0, n-1}]), a[0] == 0}, a, {n, 0, 15}] (* Vaclav Kotesovec, Apr 10 2018 *)
nmax = 15; Flatten[{0, Table[CoefficientList[TrigToExp[Expand[FunctionExpand[ Table[FullSimplify[Sum[(-1)^j/(Binomial[2*j, j]*2^j*(2*j + (2*m - 1))), {j, 0, Infinity}]]*(-1)^m*Product[(2*k - 1)^2, {k, 0, m - 1}], {m, 1, nmax}]]]], Log[2]][[n, 1]], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 10 2018 *)

Prog

(PARI) a=vector(20); a[1]=0; for(n=2, #a, a[n]=16*(n-1)*((2*n-3)*a[n-1]+(((-1)^n)/9)*prod(k=0, n-1, (2*k-3)^2))); concat(0, a) \\ Altug Alkan, Apr 09 2018

Crossrefs

Cf. A302115.

Sequence in context: A193128 A223101 A086815 * A294163 A180504 A178024

Adjacent sequences: A302113 A302114 A302115 * A302117 A302118 A302119

Keyword

nonn

Author

Detlef Meya, Apr 01 2018

Extensions

More terms from Altug Alkan, Apr 09 2018

Status

approved