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A287029 Row sums of A286800. 4
1, 3, 13, 147, 1965, 30979, 559357, 11289219, 250794109, 6066778627, 158533572861, 4447703062787, 133309656009469, 4251322261512195, 143749952968507389, 5137921526511802371, 193589838004887201789, 7670544451820808601603, 318892867844484240154621, 13881730766388536085356547 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 1..200

Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.

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..floor((2*n-1)/3)} A286800(n,k) for n>=1.

a(n) ~ 4*exp(-7/2)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 15/(8*n) - 503/(128*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 21 2017

EXAMPLE

A(x) = x + 3*x^2 + 13*x^3 + 147*x^4 + 1965*x^5 + 30979*x^6 + ...

MATHEMATICA

terms = 20; y[_, _] = 0; Do[y[x_, t_] = (1/(-1 + y[x, t])) x (-1 - y[x, t]^2 - 2 y[x, t] (-1 + D[y[x, t], x]) + t x (-1 + y[x, t]) (2 (-1 + y[x, t])^2 + (x (-1 + y[x, t]) + y[x, t]) D[y[x, t], x])) + O[x]^n // Normal // Simplify, {n, terms+1}];

Total[CoefficientList[#, t]]& /@ CoefficientList[y[x, t], x] // Rest

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;

};

A286798_ser(N, t='t) = {

  my(v = A286795_ser(N, t)); subst(v, 'x, serreverse(x/(1-x*t*v)));

};

A286800_ser(N, t='t) = {

  my(v = A286798_ser(N, t)); 1-1/subst(v, 'x, serreverse(x*v^2));

};

A287029_ser(N) = A286800_ser(N+1, 1);

Vec(A287029_ser(20))

CROSSREFS

Cf. A049464, A286799, A286800, A287039.

Sequence in context: A041591 A001150 A108554 * A317074 A230036 A014376

Adjacent sequences:  A287026 A287027 A287028 * A287030 A287031 A287032

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, May 22 2017

STATUS

approved

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Last modified October 13 19:47 EDT 2019. Contains 327981 sequences. (Running on oeis4.)