login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. Polynomes orthogonaux et transformations integrales. 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/(1-3x/(1-4x/(1-5x/(1-6x/(1-7x/(1-8x/(...))))))) (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, B(x) 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: A192461 A199682 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 11 07:40 EST 2019. Contains 329914 sequences. (Running on oeis4.)