A302945 a(n) = 4*(2*n+1)(n*a(n-1) + (-1)^(n-1)*((n-1)!)^2), with a(0) = 0, n > 0.

0, 12, 460, 38752, 5578992, 1227403584, 382949169408, 160838682255360, 87496241419607040, 59847429254564597760, 50271840562772982988800, 50875102650737732213145600, 61050123180725943563550720000, 85714372945764004556767887360000, 139200141663916245408777987686400000

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..212

Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 14, (3.89) - (3.93).

Formula

a(n-1) = f3(n)*(-1)^n*((n-1)!)^2, where f3(n) corresponds to the z values such that Sum_{k>=0} (-1)^k/(binomial(2*k,k)*2^k*(k+n)) = x*log(2) + y*(log(2))^2 + z. (See examples for connection with a(n) in terms of material at Links section.)

f2(n) corresponds to the y values, so f2(n) = (-1)^n*2^n*((2*n-1)!/((n-1)!)^2).

(2*n + 1)*a(n + 1) - (14*n^3 + 45*n^2 + 44*n + 12)*a(n) -4*n^3*(4*n^2 + 8*n + 3)*a(n - 1) = 0. - Robert Israel, Dec 02 2020

Example

Examples ((3.89) - (3.93)) at page 14 in Links section as follows, respectively.
For n=1, f3(1) = 0, so a(0) = 0.
For n=2, f3(2) = 12, so a(1) = 12.
For n=3, f3(3) = -115, so a(2) = 460.
For n=4, f3(4) = 9688/9, so a(3) = 38752.
For n=5, f3(5) = -38743/4, so a(4) = 5578992.

Maple

f:= gfun:-rectoproc({(2*n + 1)*a(n + 1) - (14*n^3 + 45*n^2 + 44*n + 12)*a(n) -4*n^3*(4*n^2 + 8*n + 3)*a(n - 1) = 0, a(0)=0, a(1)=12}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Dec 02 2020

Mathematica

RecurrenceTable[{a[n] == 4*(2*n+1)*(n*a[n-1] + (-1)^(n-1)*((n-1)!)^2), a[0] == 0}, a, {n, 0, 20}] (* Altug Alkan, Apr 16 2018 *)
nmax = 15; Table[CoefficientList[TrigToExp[Expand[FunctionExpand[Table[ Sum[(-1)^j/(Binomial[2*j, j]*2^j*(j + m)), {j, 0, Infinity}], {m, 1, nmax}]]]], Log[2]][[n, 1]]*(-1)^n*(n - 1)!^2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 18 2018 *)

Prog

(PARI) a=vector(20); a[1]=12; for(n=2, #a, a[n]=4*(2*n+1)*(n*a[n-1] + (-1)^(n-1)*((n-1)!)^2)); concat(0, a) \\

Crossrefs

Cf. A302944.

Sequence in context: A112363 A221032 A221496 * A089956 A178217 A262584

Adjacent sequences: A302942 A302943 A302944 * A302946 A302947 A302948

Keyword

nonn

Author

Detlef Meya, Apr 16 2018

Status

approved