OFFSET
0,2
LINKS
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-1)*f1(n-1)*5*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*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: (2*n - 3)*a(n) = 3*(2*n - 1)^2*a(n-1) + 8*(n-2)*(2*n - 3)*(2*n - 1)*a(n-2).
a(n) ~ sqrt(5*Pi) * 2^(3*n - 1/2) * n^(n + 1/2) / exp(n). (End)
EXAMPLE
Examples ((3.48) - (3.52)) at page 11 in Links section as follows, respectively.
For n=0, f1(0) = 4/5, so a(1) = 4.
For n=1, f1(1) = -36/5, so a(2) = 108.
For n=2, f1(2) = 572/15, so a(3) = 2860.
For n=3, f1(3) = -916/5, so a(4) = 96180.
For n=4, f1(4) = 29308/35, so a(5) = 3956580.
MATHEMATICA
RecurrenceTable[{a[m+1] == (4/(2*m - 1))*(2*m*(2*m + 1)*a[m] + (-1)^(m + 1) * Product[2*k + 1, {k, 0, m}]), a[0] == 0}, a, {m, 0, 15}] (* Vaclav Kotesovec, Apr 11 2018 *)
nmax = 15; Flatten[{0, Table[CoefficientList[1/Sqrt[5] * TrigToExp[Expand[ FunctionExpand[Table[FullSimplify[Sum[(-1)^j/(Binomial[2*j, j]*(2*j + (2*m + 1))), {j, 0, Infinity}]]*(-1)^m * 5 * Product[2*k + 1, {k, 0, m}], {m, 0, nmax}]]]], Log[1/2 + Sqrt[5]/2]][[n, 2]], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 11 2018 *)
PROG
(PARI) a=vector(20); a[1]=4; for(n=2, #a, a[n]=(4/(2*n-3))*(2*(n-1)*(2*n-1)*a[n-1]+((-1)^n)*prod(k=0, n-1, (2*k+1)))); concat(0, a) \\ Altug Alkan, Apr 01 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Detlef Meya, Apr 01 2018
EXTENSIONS
More terms from Altug Alkan, Apr 01 2018
STATUS
approved