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A167872 A sequence of moments connected with Feynman numbers (A000698): Half the number of Feynman diagrams of order 2(n+1), for the electron self-energy in quantum electrodynamics (QED), i.e., all proper diagrams including Furry vanishing diagrams (those that vanish in 4-dimensional QED because of Furry theorem). 15
1, 3, 21, 207, 2529, 36243, 591381, 10786527, 217179009, 4782674403, 114370025301, 2952426526767, 81864375589089, 2427523337157363, 76683680366193621, 2571609710380950207, 91265370849151405569, 3417956847888948899523 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is the moment of order 2*n of the probability density function defined by rho(x)=sqrt(Pi/2)*exp(-x^2/2)/((xphi(x)+1)^2+Pi^2*x^2*exp(-x^2)), where phi(x)=int(t*log(abs(x-t))*exp(-t^2/2),t=-infinity..infinity).

a(n) = A115974(n)/2, see comments in A115974. See also A000698, A005411, A005412. - Robert Coquereaux, Sep 14 2014

REFERENCES

Roland Groux. Polynômes orthogonaux et transformations intégrales. Cepadues. 2008. pages 195..206.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..400

Trinh Khanh Duy and Tomoyuki Shirai, The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles, arXiv preprint, 2015.

Adrian Ocneanu, On the inner structure of a permutation: bicolored partitions and Eulerians, trees and primitives; arXiv preprint arXiv:1304.1263, 2013

Wikipedia, Feynman diagram

FORMULA

Sum_(a(n)/z^(2n+1),n=0..infinity) = (1/2)*(z-S(z)/(z*S(z)-1)) with S(z) = Sum_((2*n)!/(2^n*n!*z^(2*n+1)), n=0..infinity).

a(n) = (2*n - 1) * a(n-1) + 2 * Sum_{k=1..n} a(k-1) * a(n-k) if n>0. - Michael Somos, Jul 23 2011

a(0)=1; for n > 0 a(n) = A000698(n+2)/2-Sum_(A000698(n+1-k)*a(k), k=0..n-1).

G.f.: 1/(1-3*x/(1-4*x/(1-5*x/(1-6*x/(1-7*x/(1-8*x/(...))))))) (continued fraction). - Philippe Deléham, Nov 20 2011

G.f.: 1/Q(0), where Q(k) = 1 - x*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 20 2013

Let A(x) be the g.f. of A127059 and B(x) be the g.f. of A167872. Then A(x) = (1 - 1/B(x))/x.

G.f.: 1/Q(0), where Q(k) = 1 - x*(2*k+3)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013

G.f.: G(0)/2, where G(k) = 1  + 1/(1 - (2*k+3)*x/((2*k+2)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013

G.f.: G(0), where G(k) = 1 - x*(k+3)/(x*(k+3) - 1/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Aug 05 2013

a(n) ~ 2^(n + 3/2) * n^(n+2) / exp(n). - Vaclav Kotesovec, Jan 02 2019

EXAMPLE

G.f. = 1 + 3*x + 21*x^2 + 207*x^3 + 2529*x^4 + 36243*x^5 + 591381*x^6 + ...

MATHEMATICA

(* f = A000698 *) f[n_] := f[n] = (2*n - 1)!! - Sum[f[n - k]*(2*k - 1)!!, {k, 1, n - 1}]; a[n_] := a[n] = f[n + 2]/2 - Sum[f[n + 1 - k]*a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 03 2013, from 3rd formula *)

PROG

(PARI) {a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 3) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 23 2011 */

CROSSREFS

Cf. A000698, A115974, A005411, A005412, A001147.

Sequence in context: A199682 A348912 A309638 * A192314 A242635 A136223

Adjacent sequences:  A167869 A167870 A167871 * A167873 A167874 A167875

KEYWORD

nonn

AUTHOR

Groux Roland, Nov 14 2009

EXTENSIONS

Name clarified from Robert Coquereaux, Sep 14 2014

STATUS

approved

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Last modified December 1 14:10 EST 2021. Contains 349430 sequences. (Running on oeis4.)