A287039 Row sums of A286782.

1, 1, 9, 100, 1323, 20088, 342430, 6461208, 133618275, 3006094768, 73139285178, 1914937983000, 53720914023150, 1608612191370000, 51235727245542684, 1730349877484075120, 61783682196714238755, 2326122843950925857376, 92117389831885545623650, 3828375469597215729851928

Offset: 0

Author

Gheorghe Coserea, May 18 2017

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..302
• Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
• Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.
• Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Crossrefs

Cf. A000699, A049464, A286782, A287029.

Programs

Mathematica

max = 21; (* B(x) is A000699(x) *) B[_] = 0;
Do[B[x_] = x + x^2 D[B[x]^2/x, x] + O[x]^max // Normal, max];
Join[{1}, Drop[CoefficientList[(1-x/B[x])/x + O[x]^max, x], -2] Table[2n-1, {n, max-2}]] (* Jean-François Alcover, Oct 25 2018, from PARI *)

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);
A287039_seq(N) = {
  my(s = A286794_seq(N));
  concat(1, vector(#s, n, (2*n-1)*s[n]));
};
A287039_seq(19)

Formula

a(n) ~ 4*exp(-1)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 19/(8*n) - 23/(128*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 21 2017