OFFSET
0,1
COMMENTS
Ei(.) is the exponential integral.
This is the case k=4 in the family a(n,k) = (1/k!)*( (n+k+2)!-(k+1)*(n+k+1)! -Sum_{i=0..n-1} (n+k-i)!*a(i,k) ). The values k = 0 to 3 are represented by A003319, A111537, A111546, and A111556.
a(n,k) is the moment of order n for the density k!*x^k*exp(-x)/((x^k*exp(-x)*Ei(x) - Pk(x))^2 + Pi^2*x^(2*k)*exp(-2*x)) on the interval 0..infinity with polynomials Pk(x) = Sum_{i=0..k-1} (k-1-i)!*x^i.
REFERENCES
R. Groux, Polynômes orthogonaux et transformations intégrales, Cépadués, 2008, 125-129.
FORMULA
a(n) = (1/24)*( (n+6)! - 5*(n+5)! - Sum_{i=0..n-1} (n+4-i)!*a(i) ).
a(n) = 5*A111532(n+1) (conjecture). - R. J. Mathar, Dec 14 2010
G.f.: 1/x/Q(0) - 1/x, where Q(k) = 1 - 3*x + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
G.f.: (1-x-2/G(0))/x^2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k-1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
G.f.: 1/x^2 - 5/x - 2/(x^2*G(0)), where G(k) = 1 + 1/(1 - x*(k+5)/(x*(k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
PROG
(PARI) a(n)=if(n==0, 5, (1/24)*( (n+6)! -5*(n+5)! -sum(i=0, n-1, (n+4-i)!*a(i) ) ) ); \\ Joerg Arndt, May 04 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Groux Roland, Dec 10 2010
STATUS
approved