A286797 - OEIS

%I #17 Mar 08 2022 14:17:24

%S 1,2,10,82,898,12018,187626,3323682,65607682,1424967394,33736908874,

%T 864372576626,23825543471234,703074672632018,22118247888976170,

%U 739081808704195650,26146116129400483842,976382058777174451650,38386296866727499728522,1584986693941237056394386

%N Row sums of A286796.

%H Gheorghe Coserea, <a href="/A286797/b286797.txt">Table of n, a(n) for n = 0..200</a>

%H Luca G. Molinari, Nicola Manini, <a href="https://arxiv.org/abs/cond-mat/0512342">Enumeration of many-body skeleton diagrams</a>, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

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

%F a(n) ~ 2^(n + 5/2) * n^(n+2) / exp(n+2). - _Vaclav Kotesovec_, Mar 08 2022

%t 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]];

%t a[n_] := CoefficientList[SeriesCoefficient[y0[x, t]/(1 - x*t*y0[x, t]), {x, 0, n}], t] // Total;

%t Table[a[n], {n, 0, max-1}] (* _Jean-François Alcover_, May 24 2017, adapted from PARI *)

%o (PARI)

%o A286795_ser(N, t='t) = {

%o   my(x='x+O('x^N), y0=1, y1=0, n=1);

%o   while(n++,

%o     y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');

%o     y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;

%o };

%o A286796_ser(N,t='t) = my(v=A286795_ser(N,t)); v/(1-x*t*v);

%o Vec(A286796_ser(20,1))

%o (PARI)

%o A049464_ser(N) = {  \\ for A049464(0)=0

%o   my(s=Ser(concat(1, vector(N+1, n, (2*n)!/(2^n*n!)))), g=(1/s - 1/s^2)/x);

%o   1 - 1/subst(g, 'x, serreverse(x*g^2*s^2));

%o };

%o A286797_ser(N) = my(q=A049464_ser(N)); q/(x-x*q);

%o Vec(A286797_ser(20))

%Y Cf. A286796.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, May 21 2017