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A286794 Row sums of A286781. 3
1, 3, 20, 189, 2232, 31130, 497016, 8907885, 176829104, 3849436062, 91187523000, 2335691914050, 64344487654800, 1897619527612692, 59667237154623280, 1993022006345620605, 70488571028815935072, 2631925423768158446390, 103469607286411235941944, 4272438866376100717458486 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
E.Z.Kuchinskii, M.V.Sadovskii, Combinatorics of Feynman diagrams for the problems with gaussian random field, arXiv:cond-mat/9706062 [cond-mat.dis-nn], 1997.
Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.
FORMULA
a(n) = Sum_{k=0..n} A286781(n,k).
A(x) = (1-x/A000699(x))/x, A208975(x) = 1 + x*A(-x).
a(n) ~ 4*exp(-1)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 3/(8*n) - 215/(128*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 23 2017
EXAMPLE
A(x) = 1 + 3*x + 20*x^2 + 189*x^3 + 2232*x^4 + 31130*x^5 + ...
MATHEMATICA
max = 22; (* B(x) is A000699(x) *) B[_] = 0;
Do[B[x_] = x + x^2 D[B[x]^2/x, x] + O[x]^max // Normal, max];
A[x_] = (1 - x/B[x])/x + O[x]^max;
Drop[CoefficientList[A[x], x], -2] (* Jean-François Alcover, Oct 25 2018 *)
PROG
(PARI)
A286781_ser(N, t='t) = {
my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
while(n++,
y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
if (y1 == y0, break()); y0 = y1; );
y0;
};
Vec(A286781_ser(20, 1))
(PARI)
A000699_seq(N) = {
my(a = vector(N)); a[1] = 1;
for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
A286794_seq(N) = Vec((1-1/Ser(A000699_seq(N+1)))/x);
A286794_seq(20)
CROSSREFS
Sequence in context: A073767 A226349 A208975 * A176043 A108206 A349959
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, May 16 2017
STATUS
approved

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Last modified July 25 03:25 EDT 2024. Contains 374586 sequences. (Running on oeis4.)