A286797 Row sums of A286796.

1, 2, 10, 82, 898, 12018, 187626, 3323682, 65607682, 1424967394, 33736908874, 864372576626, 23825543471234, 703074672632018, 22118247888976170, 739081808704195650, 26146116129400483842, 976382058777174451650, 38386296866727499728522, 1584986693941237056394386

Offset: 0

Author

Gheorghe Coserea, May 21 2017

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..200
• Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Crossrefs

Cf. A286796.

Programs

Mathematica

max = 20; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = (1 + x*(1 + 2*t + x*t^2)*y0[x, t]^2 + t*(1 - t)*x^2*y0[x, t]^3 + 2*x^2*y0[x, t]*D[y0[x, t], x])/(1 + 2*x*t) + O[x]^n // Normal // Simplify; y0[x_, t_] = y1[x, t]];
a[n_] := CoefficientList[SeriesCoefficient[y0[x, t]/(1 - x*t*y0[x, t]), {x, 0, n}], t] // Total;
Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, May 24 2017, adapted from PARI *)

PARI

A286795_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++,
    y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
    y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
};
A286796_ser(N,t='t) = my(v=A286795_ser(N,t)); v/(1-x*t*v);
Vec(A286796_ser(20,1))

PARI

A049464_ser(N) = {  \\ for A049464(0)=0
  my(s=Ser(concat(1, vector(N+1, n, (2*n)!/(2^n*n!)))), g=(1/s - 1/s^2)/x);
  1 - 1/subst(g, 'x, serreverse(x*g^2*s^2));
};
A286797_ser(N) = my(q=A049464_ser(N)); q/(x-x*q);
Vec(A286797_ser(20))

Formula

a(n) = Sum_{k=0..n} A286796(n,k).

a(n) ~ 2^(n + 5/2) * n^(n+2) / exp(n+2). - Vaclav Kotesovec, Mar 08 2022