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%I #21 Oct 24 2018 03:30:56
%S 1,3,13,147,1965,30979,559357,11289219,250794109,6066778627,
%T 158533572861,4447703062787,133309656009469,4251322261512195,
%U 143749952968507389,5137921526511802371,193589838004887201789,7670544451820808601603,318892867844484240154621,13881730766388536085356547
%N Row sums of A286800.
%H Gheorghe Coserea, <a href="/A287029/b287029.txt">Table of n, a(n) for n = 1..200</a>
%H Michael Borinsky, <a href="https://arxiv.org/abs/1703.00840">Renormalized asymptotic enumeration of Feynman diagrams</a>, arXiv:1703.00840 [hep-th], 2017.
%H Luca G. Molinari, Nicola Manini, <a href="https://arxiv.org/abs/cond-mat/0512342">Enumeration of many-body skeleton diagrams</a>, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.
%F a(n) = Sum_{k=0..floor((2*n-1)/3)} A286800(n,k) for n>=1.
%F a(n) ~ 4*exp(-7/2)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 15/(8*n) - 503/(128*n^2) + O(1/n^3)). (see Borinsky link) - _Gheorghe Coserea_, Oct 21 2017
%e A(x) = x + 3*x^2 + 13*x^3 + 147*x^4 + 1965*x^5 + 30979*x^6 + ...
%t terms = 20; y[_, _] = 0; Do[y[x_, t_] = (1/(-1 + y[x, t])) x (-1 - y[x, t]^2 - 2 y[x, t] (-1 + D[y[x, t], x]) + t x (-1 + y[x, t]) (2 (-1 + y[x, t])^2 + (x (-1 + y[x, t]) + y[x, t]) D[y[x, t], x])) + O[x]^n // Normal // Simplify, {n, terms+1}];
%t Total[CoefficientList[#, t]]& /@ CoefficientList[y[x, t], x] // Rest
%o (PARI)
%o A286795_ser(N, t='t) = {
%o my(x='x+O('x^N), y0=1, y1=0, n=1);
%o while(n++,
%o y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
%o y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
%o };
%o A286798_ser(N,t='t) = {
%o my(v = A286795_ser(N,t)); subst(v, 'x, serreverse(x/(1-x*t*v)));
%o };
%o A286800_ser(N, t='t) = {
%o my(v = A286798_ser(N,t)); 1-1/subst(v, 'x, serreverse(x*v^2));
%o };
%o A287029_ser(N) = A286800_ser(N+1, 1);
%o Vec(A287029_ser(20))
%Y Cf. A049464, A286799, A286800, A287039.
%K nonn
%O 1,2
%A _Gheorghe Coserea_, May 22 2017
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