A302114 a(n) = 8*(n-1)*(2*n-3)*a(n-1) + ((-1)^n)*(n-1)*Product_{k=0..n-3} (2*k+1)^2 with a(0) = 0.

0, 0, 1, 46, 5547, 1241628, 447041205, 236032398090, 171832342201695, 164958902421881400, 201909733543855989225, 306902783112141390315750, 567156347906609067457417875, 1252281213909270325492296700500, 3255931157464155279716218995508125

Offset

0,4

Links

Table of n, a(n) for n=0..14.

Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 11, (3.48) - (3.52).

Formula

a(n) = (-1)^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)^k/(binomial(2*k,k)*(2*k+(2*n+1))) = x*sqrt(5)*log((1+sqrt(5))/2) + y. (See examples for connection with a(n) in terms of material at Links section).

From Vaclav Kotesovec, Nov 22 2024: (Start)

Recurrence: (n-2)*a(n) = (n-1)*(12*n^2 - 36*n + 23)*a(n-1) + 8*(n-2)*(n-1)*(2*n - 5)^3*a(n-2).

a(n) ~ sqrt(Pi) * log(2 + sqrt(5)) * n^(2*n - 3/2) * 2^(4*n - 5) / (3*exp(2*n)). (End)

Example

Examples ((3.48) - (3.52)) at page 11 in Links section as follows, respectively.
For n=0, f2(0) = 0, so a(1) = 0.
For n=1, f2(1) = 8, so a(2) = 1.
For n=2, f2(2) = -368/9, so a(3) = 46.
For n=3, f2(3) = 14792/75, so a(4) = 5547.
For n=4, f2(4) = -3311008/3675, so a(5) = 1241628.

Mathematica

RecurrenceTable[{a[m+1] == 8*m*(2*m - 1)*a[m] + (-1)^(m + 1)*m * Product[(2*k + 1)^2, {k, 0, m - 2}], a[0] == 0}, a, {m, 0, 15}] (* Vaclav Kotesovec, Apr 11 2018 *)
nmax = 15; Flatten[{0, 0, Table[CoefficientList[TrigToExp[Expand[FunctionExpand[ Table[FullSimplify[Sum[(-1)^j/(Binomial[2*j, j]*(2*j + (2*m + 1))), {j, 0, Infinity}]]*(-1)^(m + 1)/8 * Product[(2*k + 1)^2, {k, 0, m - 1}], {m, 1, nmax}]]]], Log[1/2 + Sqrt[5]/2]][[n, 1]], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 11 2018 *)

Prog

(PARI) a=vector(20); a[1]=0; for(n=2, #a, a[n]=8*(n-1)*(2*n-3)*a[n-1]+(-1)^n*(n-1)*prod(k=0, n-3, (2*k+1)^2)); concat(0, a) \\ Altug Alkan, Apr 01 2018

Crossrefs

Cf. A302113.

Sequence in context: A223884 A223891 A243648 * A159380 A134793 A177639

Adjacent sequences: A302111 A302112 A302113 * A302115 A302116 A302117

Keyword

nonn

Author

Detlef Meya, Apr 01 2018

Extensions

More terms from Altug Alkan, Apr 01 2018

Status

approved