A226260 Numerators of mass formula for connected vacuum graphs on 2n nodes for a phi^3 field theory.

1, 5, 5, 1105, 565, 82825, 19675, 1282031525, 80727925, 1683480621875, 13209845125, 2239646759308375, 19739117098375, 6320791709083309375, 32468078556378125, 38362676768845045751875, 281365778405032973125, 2824650747089425586152484375, 776632157034116712734375

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Carl. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304052, 1993. See Eq. 15.

Carl. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, Journal of Mathematical Physics 35, 1994, 364-367.

Formula

Let V(n) = (3*n - 1)!!/(3!^n*n!), and c(n) = V(2*n) - (1/n)*Sum_{j=0..n-1} j*c(j)*V(2*(n-j)), c(0) = 1. Then a(n) = numerator of c(n). - Franck Maminirina Ramaharo, Feb 04 2019

Example

1, 5/24, 5/16, 1105/1152, 565/128, 82825/3072, 19675/96, 1282031525/688128, 80727925/4096, ...

Prog

(Maxima)
c_list : [1]$
V(n) := if n = 0 then 1 else (3*n - 1)!!/(3!^n*n!)$
c(n) := V(2*n) - 1/n*sum(j*c_list[j + 1]*V(2*(n - j)), j , 0 , n - 1)$
for i:1 thru 50 do c_list : append(c_list, [c(i)])$
map(num, c_list); /* Franck Maminirina Ramaharo, Feb 04 2019 */

Crossrefs

Cf. A226261, A226256, A226256-A226259, A225697, A225698.

Sequence in context: A213145 A195567 A051716 * A102060 A102058 A231409

Adjacent sequences: A226257 A226258 A226259 * A226261 A226262 A226263

Keyword

nonn,frac

Author

N. J. A. Sloane, Jun 02 2013

Extensions

More terms from Franck Maminirina Ramaharo, Feb 04 2019

Status

approved