A303108 a(n) = (2*n-1)*a(n-1) - (n-2)!, with a(1) = 2, n > 1.

2, 5, 24, 166, 1488, 16344, 212352, 3184560, 54132480, 1028476800, 21597649920, 496742319360, 12418518067200, 335299508812800, 9723679528550400, 301433978206771200, 9947319973149081600, 348156178137427968000, 12881778235397406720000, 502389344778125156352000

Offset

1,1

Links

Chai Wah Wu, Table of n, a(n) for n = 1..404

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

Formula

a(n) = f1(n)*2*(n-1)!, where f1(n) corresponds to the x 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) = 3*(n-1)*a(n-1) - (2n-3)*(n-2)*a(n-2) for n > 2. - Chai Wah Wu, Apr 20 2018

Example

Examples ((3.94) - (3.98)) at page 15 in Links section as follows, respectively.
For n=1, f1(1) = 1, so a(1) = 2.
For n=2, f1(2) = 5/2, so a(2) = 5.
For n=3, f1(3) = 6, so a(3) = 24.
For n=4, f1(4) = 83/6, so a(4) = 166.
For n=5, f1(5) = 31, so a(5) = 1488.

Mathematica

RecurrenceTable[{a[n] == (2*n-1)*a[n-1] - (n-2)!, a[1] == 2}, a, {n, 1, 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, 2]]*2*(n-1)!, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 20 2018 *)

Prog

(PARI) a=vector(20); a[1]=2; for(n=2, #a, a[n]=(2*n-1)*a[n-1] - (n-2)!); a
(Python)
A303108_list = [2, 5]
for n in range(3, 501):
    A303108_list.append(3*(n-1)*A303108_list[-1]-(2*n-3)*(n-2)*A303108_list[-2]) # Chai Wah Wu, Apr 20 2018
(Magma) I:=[2, 5]; [n le 2 select I[n] else 3*(n-1)*Self(n-1)- (2*n-3)*(n-2)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 20 2018

Crossrefs

Cf. A303109.

Sequence in context: A086635 A020124 A012249 * A331035 A052808 A137157

Adjacent sequences: A303105 A303106 A303107 * A303109 A303110 A303111

Keyword

nonn

Author

Detlef Meya, Apr 18 2018

Status

approved