A177354 a(n) is the moment of order n for the density measure 24*x^4*exp(-x)/( (x^4*exp(-x)*Ei(x) - x^3 - x^2 - 2*x - 6)^2 + Pi^2*x^8*exp(-2*x) ) over the interval 0..infinity.

5, 35, 305, 3095, 35225, 439775, 5939225, 85961375, 1324702025, 21632195375, 372965377625, 6769644905375, 129049505347625, 2578419996023375, 53898389265685625, 1176832196718869375, 26798832693476455625, 635575680349115699375, 15677971277701873945625, 401729457433222058609375

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.

Links

Table of n, a(n) for n=0..19.

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

Sequence in context: A027392 A291813 A346765 * A253096 A305964 A226739

Adjacent sequences: A177351 A177352 A177353 * A177355 A177356 A177357

Keyword

nonn

Author

Groux Roland, Dec 10 2010

Status

approved