A226259 Denominators of mass formula for connected vacuum graphs on n nodes for a phi^4 field theory.

1, 8, 12, 96, 72, 960, 324, 96768, 1296, 82944, 1620, 3649536, 17496, 77635584, 6804, 159252480, 23328, 9746251776, 708588, 392143306752, 2361960, 866843099136, 7794468, 22785590034432, 25509168, 445805022412800, 82904796, 213986410758144, 29760696

Offset

0,2

Links

Table of n, a(n) for n=0..28.

Carl M. Bender and Kimball A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304052, 1993. See Eq. 11.

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

Formula

Let V(n) = (4*n - 1)!!/(4!^n*n!) = A225697(n)/A225698(n), and c(n) = V(n) - (1/n)*Sum_{j=0..n-1} j*c(j)*V(n-j), c(0) = 1. Then a(n) = denominator of c(n). - Franck Maminirina Ramaharo, Feb 04 2019

Example

1, 1/8, 1/12, 11/96, 17/72, 619/960, 709/324, ...

Prog

(Maxima)
c_list : [1]$
V(n) := if n = 0 then 1 else (4*n - 1)!!/(4!^n*n!)$
c(n) := V(n) - 1/n*sum(j*c_list[j + 1]*V(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. A226258, A226256, A226257, A226260, A226261, A225697, A225698.

Sequence in context: A216711 A137232 A147764 * A162466 A098664 A266800

Adjacent sequences: A226256 A226257 A226258 * A226260 A226261 A226262

Keyword

nonn,frac

Author

N. J. A. Sloane, Jun 02 2013

Extensions

More terms from Franck Maminirina Ramaharo, Feb 04 2019

Status

approved