A303109 a(n) = n*(2*n-1)*a(n-1) + ((n-1)!)^2, with a(0) = 0, n > 0.

0, 1, 7, 109, 3088, 139536, 9223776, 839882016, 100811243520, 15425745960960, 2931023414476800, 677079576933580800, 186875556584590540800, 60734785332524728320000, 22957787631482390937600000, 9986645219749296609853440000, 4953377739007903842686730240000, 2778845349346570753142308208640000

Offset

0,3

Links

Chai Wah Wu, Table of n, a(n) for n = 0..238

Miroslav Kureš, A Note on the Remarkable Expression of the Number 8/Pi^2 That the Ramanujan Machine Discovered, Mathematical Intelligencer (2021).

Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 15, (3.94) - (3.98).

Formula

a(n-1) = f3(n) * ((n-1)!)^2 / (2*n-1), where f3(n) corresponds to the z values such that Sum_{k>=0} 2^k/(binomial(2*k,k)*(k+n))) = x*Pi - y*Pi^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) = (1/2^(n+2))*((2*n-1)!/((n-1)!)^2).

a(n) = (3n^2-3n+1)*a(n-1) - (n-1)^3*(2n-3)*a(n-2) for n > 1. - Chai Wah Wu, Apr 20 2018

a(n) = (2*n)!/2^n * Sum_{k=1..n} 2^k*(k-1)!^2/(2*k)! [Kureš]. - Alois P. Heinz, Jan 26 2022

Example

Examples ((3.94) - (3.98)) at page 15 in Links section as follows, respectively.
For n=1, f3(1) = 0, so a(0) = 0.
For n=2, f3(2) = 3, so a(1) = 1.
For n=3, f3(3) = 35/4, so a(2) = 7.
For n=4, f3(4) = 763/36, so a(3) = 109.
For n=5, f3(5) = 193/4, so a(4) = 3088.

Mathematica

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

Prog

(PARI) a=vector(20); a[1]=1; for(n=2, #a, a[n]=n*(2*n-1)*a[n-1] + ((n-1)!)^2); concat(0, a)
(Python)
A303109_list = [0, 1]
for n in range(2, 501):
    A303109_list.append((3*n*(n-1)+1)*A303109_list[-1]-(2*n-3)*(n-1)**3*A303109_list[-2]) # Chai Wah Wu, Apr 20 2018

Crossrefs

Cf. A303108.

Sequence in context: A116875 A357395 A371317 * A101924 A171193 A357393

Adjacent sequences: A303106 A303107 A303108 * A303110 A303111 A303112

Keyword

nonn

Author

Detlef Meya, Apr 18 2018

Status

approved