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URL: https://oeis.org/A287029

⇱ A287029 - OEIS

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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
OFFSET
1,2
LINKS
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
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, May 22 2017
STATUS
approved