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

0,2

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, 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.

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

Example

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

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 *)

Prog

(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)

Crossrefs

Cf. A208975, A286781.

Sequence in context: A073767 A226349 A208975 * A176043 A108206 A349959

Adjacent sequences: A286791 A286792 A286793 * A286795 A286796 A286797

Keyword

nonn

Author

Gheorghe Coserea, May 16 2017

Status

approved