A226261 Denominators of mass formula for connected vacuum graphs on 2n nodes for a phi^3 field theory.

1, 24, 16, 1152, 128, 3072, 96, 688128, 4096, 7077888, 4096, 46137344, 24576, 436207616, 114688, 6442450944, 2097152, 876173328384, 9437184, 15668040695808, 8388608, 138538465099776, 1441792, 4855443348258816, 201326592, 1688849860263936, 872415232

Offset

0,2

Links

Table of n, a(n) for n=0..26.

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) = denominator 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(denom, c_list); /* Franck Maminirina Ramaharo, Feb 04 2019 */

Crossrefs

Cf. A226260, A226256-A226259, A225697, A225698.

Sequence in context: A033826 A033824 A040554 * A028696 A284874 A216841

Adjacent sequences: A226258 A226259 A226260 * A226262 A226263 A226264

Keyword

nonn,frac

Author

N. J. A. Sloane, Jun 02 2013

Extensions

More terms from Franck Maminirina Ramaharo, Feb 04 2019

Status

approved