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A286794 Row sums of A286781.

%I A286794 #42 May 27 2025 06:27:10
%S A286794 1,3,20,189,2232,31130,497016,8907885,176829104,3849436062,
%T A286794 91187523000,2335691914050,64344487654800,1897619527612692,
%U A286794 59667237154623280,1993022006345620605,70488571028815935072,2631925423768158446390,103469607286411235941944,4272438866376100717458486
%N A286794 Row sums of A286781.
%H A286794 Gheorghe Coserea, <a href="/A286794/b286794.txt">Table of n, a(n) for n = 0..301</a>
%H A286794 Michael Borinsky, <a href="https://arxiv.org/abs/1703.00840">Renormalized asymptotic enumeration of Feynman diagrams</a>, arXiv:1703.00840 [hep-th], 2017.
%H A286794 E. Z. Kuchinskii and M. V. Sadovskii, <a href="https://arxiv.org/abs/cond-mat/9706062">Combinatorics of Feynman diagrams for the problems with gaussian random field</a>, arXiv:cond-mat/9706062 [cond-mat.dis-nn], 1997.
%H A286794 Luca G. Molinari, <a href="https://arxiv.org/abs/cond-mat/0401500">Hedin's equations and enumeration of Feynman's diagrams</a>, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.
%F A286794 a(n) = Sum_{k=0..n} A286781(n,k).
%F A286794 A(x) = (1-x/A000699(x))/x, A208975(x) = 1 + x*A(-x).
%F A286794 a(n) ~ 4*exp(-1)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 3/(8*n) - 215/(128*n^2) + O(1/n^3)). (see Borinsky link) - _Gheorghe Coserea_, Oct 23 2017
%e A286794 A(x) = 1 + 3*x + 20*x^2 + 189*x^3 + 2232*x^4 + 31130*x^5 + ...
%t A286794 max = 22; (* B(x) is A000699(x) *) B[_] = 0;
%t A286794 Do[B[x_] = x + x^2 D[B[x]^2/x, x] + O[x]^max // Normal, max];
%t A286794 A[x_] = (1 - x/B[x])/x + O[x]^max;
%t A286794 Drop[CoefficientList[A[x], x], -2] (* _Jean-François Alcover_, Oct 25 2018 *)
%o A286794 (PARI)
%o A286794 A286781_ser(N, t='t) = {
%o A286794   my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
%o A286794   while(n++,
%o A286794     y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
%o A286794     if (y1 == y0, break()); y0 = y1;);
%o A286794   y0;
%o A286794 };
%o A286794 Vec(A286781_ser(20,1))
%o A286794 (PARI)
%o A286794 A000699_seq(N) = {
%o A286794   my(a = vector(N)); a[1] = 1;
%o A286794   for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
%o A286794 };
%o A286794 A286794_seq(N) = Vec((1-1/Ser(A000699_seq(N+1)))/x);
%o A286794 A286794_seq(20)
%Y A286794 Cf. A208975, A286781.
%K A286794 nonn
%O A286794 0,2
%A A286794 _Gheorghe Coserea_, May 16 2017