login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A111556
Column 3 of triangle A111553.
5
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.
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
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Aug 07 2005
STATUS
approved