A302115 a(n) = 16*(n-1)*a(n-1) + ((-1)^n)*(4/3)*Product_{k=0..n-1} (2*k-3) with a(0) = 0.

0, 4, 68, 2172, 104268, 6673092, 533847780, 51249383100, 5739930948780, 734711160903300, 105798407178183300, 16927745148371490300, 2979283146116001209100, 572022364054217234904900, 118980651723278449796792100, 26651665986014341131067107900

Offset

0,2

Links

Table of n, a(n) for n=0..15.

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*f1(n)*3*Product_{k=0..n-1} (2*k-1) where f1(n) corresponds to the x 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, f1(1) = 4/3, so a(1) = 4.
For n=2, f1(2) = -68/3, so a(2) = 68.
For n=3, f1(3) = 724/3, so a(3) = 2172.
For n=4, f1(4) = -34756/15, so a(4) = 104268.
For n=5, f1(5) = 2224364/105, so a(5) = 6673092.

Mathematica

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 * 3 * Product[(2*k - 1), {k, 0, m - 1}], {m, 1, nmax}]]]], Log[2]][[n, 2]], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 10 2018 *)
RecurrenceTable[{a[n] == 16*(n - 1)*a[n - 1] + (-1)^n*(4/3) * Product[(2*k - 3), {k, 0, n - 1}], a[0] == 0}, a, {n, 0, 15}] (* Vaclav Kotesovec, Apr 11 2018 *)

Prog

(PARI) a=vector(20); a[1]=4; for(n=2, #a, a[n]=16*(n-1)*a[n-1]+((-1)^n)*(4/3)*prod(k=0, n-1, (2*k-3))); concat(0, a) \\ Altug Alkan, Apr 10 2018

Crossrefs

Cf. A302116.

Sequence in context: A351027 A156470 A326288 * A009473 A009520 A009791

Adjacent sequences: A302112 A302113 A302114 * A302116 A302117 A302118

Keyword

nonn

Author

Detlef Meya, Apr 01 2018

Extensions

More terms from Vaclav Kotesovec, Apr 10 2018

Status

approved