A226259 - OEIS

%I #19 Feb 04 2019 09:04:44

%S 1,8,12,96,72,960,324,96768,1296,82944,1620,3649536,17496,77635584,

%T 6804,159252480,23328,9746251776,708588,392143306752,2361960,

%U 866843099136,7794468,22785590034432,25509168,445805022412800,82904796,213986410758144,29760696

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

%H Carl M. Bender and Kimball A. Milton, <a href="https://arxiv.org/abs/hep-th/9304052">Continued fraction as a discrete nonlinear transform</a>, arXiv:hep-th/9304052, 1993. See Eq. 11.

%H Carl M. Bender and Kimball A. Milton, <a href="https://doi.org/10.1063/1.530777">Continued fraction as a discrete nonlinear transform</a>, Journal of Mathematical Physics 35, 1994, 364-367.

%F 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

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

%o (Maxima)

%o c_list : [1]$

%o V(n) := if n = 0 then 1 else (4*n - 1)!!/(4!^n*n!)$

%o c(n) := V(n) - 1/n*sum(j*c_list[j + 1]*V(n - j), j , 0 , n - 1)$

%o for i:1 thru 50 do c_list : append(c_list, [c(i)])$

%o map(denom, c_list); /* _Franck Maminirina Ramaharo_, Feb 04 2019 */

%Y Cf. A226258, A226256, A226257, A226260, A226261, A225697, A225698.

%K nonn,frac

%O 0,2

%A _N. J. A. Sloane_, Jun 02 2013

%E More terms from _Franck Maminirina Ramaharo_, Feb 04 2019