A111556 Column 3 of triangle A111553.

1, 4, 24, 184, 1664, 17024, 192384, 2366144, 31362304, 444907264, 6720628224, 107674883584, 1823884857344, 32575705493504, 612054254936064, 12071987619713024, 249477777420304384, 5392386599983366144

Offset

0,2

Comments

Also found in column 0 of triangle A111559, which equals the matrix inverse of A111553.

Links

Table of n, a(n) for n=0..17.

Glenn Bruda, Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals, arXiv:2412.19866 [math.CO], 2024. See page 4.

Formula

G.f.: log(Sum_{n>=0} ((n+3)!/3!)*x^n) = Sum_{n>=1} a(n)*x^n/n.

a(n) = 4*A111531(n) for n>0.

From Groux Roland, Dec 10 2010: (Start)

6*a(n+1) = (n+5)! - 4*(n+4)! - Sum_{k=0..n-1} (n-k+3)!*a(k+1).

a(n+1) is the moment of order n for the density 6*x^3*exp(-x)/( (x^3*exp(-x)*Ei(x)-x^2-x-2)^2 + Pi^2*x^6*exp(-2*x) ) over the interval 0..infinity. (End)

a(n) = Sum_{k=0..n} A200659(n,k)*3^k. - Philippe Deléham, Nov 21 2011

G.f.: 1/(1-4x/(1-2x/(1-5x/(1-3x/(1-6x/(1-4x/(1-...(continued fraction). - Philippe Deléham, Nov 21 2011

G.f.: 1/Q(0), where Q(k) = 1 - 2*x + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013

G.f.: 1/x - 3 - 2/(x*G(0)), where G(k) = 1 + 1/(1 - x*(k+4)/(x*(k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013

G.f.: W(0), where W(k) = 1 - x*(k+4)/( x*(k+4) - 1/(1 - x*(k+2)/( x*(k+2) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013

Prog

(PARI) {a(n)=if(n<0, 0, (matrix(n+4, n+4, m, j, if(m==j, 1, if(m==j+1, -m+1, -(m-j-1)*polcoeff(log(sum(i=0, m, (i+3)!/3!*x^i)), m-j-1))))^-1)[n+4, 4])}

Crossrefs

Cf. A111553, A111531, A111559.

Sequence in context: A061720 A197472 A152403 * A300736 A226738 A271215

Adjacent sequences: A111553 A111554 A111555 * A111557 A111558 A111559

Keyword

nonn,changed

Author

Paul D. Hanna, Aug 07 2005

Status

approved