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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).
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%I #79 Jun 06 2022 03:39:16

%S 1,3,21,207,2529,36243,591381,10786527,217179009,4782674403,

%T 114370025301,2952426526767,81864375589089,2427523337157363,

%U 76683680366193621,2571609710380950207,91265370849151405569,3417956847888948899523

%N 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).

%C 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)/((x*phi(x)+1)^2 + Pi^2*x^2*exp(-x^2)), where phi(x) = Integral_{t=-oo..oo} t*log(abs(x-t))*exp(-t^2/2) dt.

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

%H Alois P. Heinz, <a href="/A167872/b167872.txt">Table of n, a(n) for n = 0..400</a>

%H Trinh Khanh Duy and Tomoyuki Shirai, <a href="http://arxiv.org/abs/1504.06904">The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles</a>, arXiv preprint arXiv:1504.06904 [math.SP], 2015.

%H Adrian Ocneanu, <a href="http://arxiv.org/abs/1304.1263">On the inner structure of a permutation: bicolored partitions and Eulerians, trees and primitives</a>; arXiv preprint arXiv:1304.1263 [math.CO], 2013.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Feynman_diagram">Feynman diagram</a>

%F Sum_{n>=0} a(n)/z^(2n+1) = (1/2)*(z-S(z)/(z*S(z)-1)) with S(z) = Sum_{n>=0} (2*n)!/(2^n*n!*z^(2*n+1)).

%F 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

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

%F 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

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

%F 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.

%F 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

%F 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

%F 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

%F a(n) = A115974(n)/2, see comments in A115974. See also A000698, A005411, A005412. - _Robert Coquereaux_, Sep 14 2014

%F a(n) ~ 2^(n + 3/2) * n^(n+2) / exp(n). - _Vaclav Kotesovec_, Jan 02 2019

%F G.f.: 1/(1 + x - 4*x/(1 - 3*x/(1 - 6*x/(1 - 5*x/(1 - 8*x/(1 - 7*x/(1 - ...))))))). - _Peter Bala_, May 30 2022

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

%t (* 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 *)

%t nmax = 20; CoefficientList[Series[1/(1 + x + ContinuedFractionK[-(k - (-1)^k)*x, 1, {k, 3, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 06 2022, after _Peter Bala_ *)

%o (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 */

%Y Row 3 of A321964. Cf. A000698, A115974, A005411, A005412, A001147.

%K nonn

%O 0,2

%A _Groux Roland_, Nov 14 2009

%E Name clarified from _Robert Coquereaux_, Sep 14 2014