|
%I #23 Feb 04 2019 09:09:54
%S 1,1,1,11,17,619,709,858437,54193,18639247,2197187,33152545703,
%T 1169890097,41657327595361,31722037141,6944870083473751,
%U 10192167279257,45494616421387671961,37539803774446801,249615310568664912892639,19065529984707154577
%N Numerators 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) = numerator 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(num,c_list); /* _Franck Maminirina Ramaharo_, Feb 04 2019 */
%Y Cf. A226259, A226256, A226257, A226260, A226261, A225697, A225698.
%K nonn,frac
%O 0,4
%A _N. J. A. Sloane_, Jun 02 2013
%E More terms from _Franck Maminirina Ramaharo_, Feb 04 2019
|
