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
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
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
KEYWORD
nonn
AUTHOR
Groux Roland, Nov 14 2009
EXTENSIONS
Name clarified from Robert Coquereaux, Sep 14 2014
STATUS
approved