OFFSET
0,2
LINKS
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) = 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)*(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 ((3.83) - (3.88)) at page 14 in Links section as follows, respectively.
For n=1, f3(1) = 0, so a(0) = 0.
For n=2, f3(2) = 6, so a(1) = 6.
For n=3, f3(3) = -55/2, so a(2) = 110.
For n=4, f3(4) = 1169/9, so a(3) = 4676.
For n=5, f3(5) = -4667/8, so a(4) = 336024.
MATHEMATICA
Fold[Append[#1, 2 (#2 (2 #2 + 1) #1[[#2]] + (-1)^(#2 - 1) (2 #2 + 1) ((#2 - 1)!)^2)] &, {0}, Range@ 19] (* Michael De Vlieger, Apr 14 2018 *)
nmax = 15; Table[CoefficientList[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, 1]]*(-1)^n*(n - 1)!^2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 15 2018 *)
PROG
(PARI) a=vector(20); a[1]=0; for(n=2, #a, a[n]=2*((n-1)*(2*n-1)*a[n-1] + (-1)^(n)*(2*n-1)*((n-2)!)^2)); a \\
CROSSREFS
KEYWORD
nonn
AUTHOR
Detlef Meya, Apr 14 2018
STATUS
approved