A226258 Numerators of mass formula for connected vacuum graphs on n nodes for a phi^4 field theory.

1, 1, 1, 11, 17, 619, 709, 858437, 54193, 18639247, 2197187, 33152545703, 1169890097, 41657327595361, 31722037141, 6944870083473751, 10192167279257, 45494616421387671961, 37539803774446801, 249615310568664912892639, 19065529984707154577

Offset

0,4

Links

Table of n, a(n) for n=0..20.

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) = numerator 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(num, c_list); /* Franck Maminirina Ramaharo, Feb 04 2019 */

Crossrefs

Cf. A226259, A226256, A226257, A226260, A226261, A225697, A225698.

Sequence in context: A146446 A228055 A132092 * A056705 A259744 A065706

Adjacent sequences: A226255 A226256 A226257 * A226259 A226260 A226261

Keyword

nonn,frac

Author

N. J. A. Sloane, Jun 02 2013

Extensions

More terms from Franck Maminirina Ramaharo, Feb 04 2019

Status

approved