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).

1, 3, 21, 207, 2529, 36243, 591381, 10786527, 217179009, 4782674403, 114370025301, 2952426526767, 81864375589089, 2427523337157363, 76683680366193621, 2571609710380950207, 91265370849151405569, 3417956847888948899523

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)/((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.

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 arXiv:1504.06904 [math.SP], 2015.

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

Wikipedia, Feynman diagram

Formula

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)).

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_{k=0..n-1} A000698(n+1-k)*a(k).

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

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

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

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 *)
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 *)

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

Row 3 of A321964. 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