A302865 a(n) = (4*n+2)*a(n-1) + (-1)^(n+1)*4*((n-1)!), with a(0) = 8, n > 0.

8, 52, 516, 7232, 130152, 2863440, 74448960, 2233471680, 75938016960, 2885644805760, 121197080390400, 5575065712473600, 278753285464012800, 15052677416972697600, 873055290159508377600, 54129427990238232576000, 3572542247350492652544000, 250077957314618176837632000

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..360

Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 14, (3.84) - (3.88).

Formula

a(n-1) = f1(n)*(-1)^(n-1)*5*((n-1)!), where f1(n) corresponds to the x values such that Sum_{k>=0} (-1)^k/(binomial(2*k,k)*(k+n)) = x*sqrt(5)*log(phi) + y*(log(phi))^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*4*((2*n-1)!/((n-1)!)^2).

Example

Examples ((3.84) - (3.88)) at page 14 in Links section as follows, respectively.
For n=1, f1(1) = 8/5, so a(0) = 8.
For n=2, f1(2) = -52/5, so a(1) = 52.
For n=3, f1(3) = 258/5, so a(2) = 516.
For n=4, f1(4) = -3616/15, so a(3) = 7232.
For n=5, f1(5) = 5423/5, so a(4) = 130152.

Mathematica

Fold[Append[#1, (4 #2 + 2) #1[[#2]] + (-1)^(#2 + 1) 4 ((#2 - 1)!)] &, {8}, Range@ 19] (* Michael De Vlieger, Apr 14 2018 *)
nmax = 15; Table[CoefficientList[1/Sqrt[5]*TrigToExp[Expand[FunctionExpand[ Table[Sum[(-1)^j/(Binomial[2*j, j]*(j + m)), {j, 0, Infinity}], {m, 1, nmax}]]]], Log[1/2 + Sqrt[5]/2]][[n, 2]]*(-1)^(n - 1)*5*(n - 1)!, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 15 2018 *)

Prog

(PARI) my(a=vector(20)); a[1]=8; for(n=2, #a, a[n]=(4*n-2)*a[n-1] + (-1)^(n)*4*(n-2)!); a \\
(Magma) [n le 1 select 8 else (4*n-2)*Self(n-1) + (-1)^(n)*4*Factorial(n-2): n in [1..20]]; // Vincenzo Librandi, Apr 17 2019

Crossrefs

Cf. A302866.

Sequence in context: A130153 A359126 A199700 * A116171 A303966 A304928

Adjacent sequences: A302862 A302863 A302864 * A302866 A302867 A302868

Keyword

nonn

Author

Detlef Meya, Apr 14 2018

Status

approved