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

0,2

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.

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

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 *)

Prog

(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))

Crossrefs

Cf. A286796.

Sequence in context: A174962 A062396 A218294 * A321089 A112487 A089469

Adjacent sequences: A286794 A286795 A286796 * A286798 A286799 A286800

Keyword

nonn

Author

Gheorghe Coserea, May 21 2017

Status

approved