A286794 Row sums of A286781.

1, 3, 20, 189, 2232, 31130, 497016, 8907885, 176829104, 3849436062, 91187523000, 2335691914050, 64344487654800, 1897619527612692, 59667237154623280, 1993022006345620605, 70488571028815935072, 2631925423768158446390, 103469607286411235941944, 4272438866376100717458486

Offset: 0

Author

Gheorghe Coserea, May 16 2017

Keywords

• nonn

Examples

			A(x) = 1 + 3*x + 20*x^2 + 189*x^3 + 2232*x^4 + 31130*x^5 + ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..301
• Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
• E. Z. Kuchinskii and M. V. Sadovskii, Combinatorics of Feynman diagrams for the problems with gaussian random field, arXiv:cond-mat/9706062 [cond-mat.dis-nn], 1997.
• Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.

Crossrefs

Cf. A208975, A286781.

Programs

Mathematica

max = 22; (* B(x) is A000699(x) *) B[_] = 0;
Do[B[x_] = x + x^2 D[B[x]^2/x, x] + O[x]^max // Normal, max];
A[x_] = (1 - x/B[x])/x + O[x]^max;
Drop[CoefficientList[A[x], x], -2] (* Jean-François Alcover, Oct 25 2018 *)

PARI

A286781_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
  while(n++,
    y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
    if (y1 == y0, break()); y0 = y1;);
  y0;
};
Vec(A286781_ser(20,1))

PARI

A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
A286794_seq(N) = Vec((1-1/Ser(A000699_seq(N+1)))/x);
A286794_seq(20)

Formula

a(n) = Sum_{k=0..n} A286781(n,k).

A(x) = (1-x/A000699(x))/x, A208975(x) = 1 + x*A(-x).

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