A111556 - OEIS

%I #29 Jan 06 2025 06:30:44

%S 1,4,24,184,1664,17024,192384,2366144,31362304,444907264,6720628224,

%T 107674883584,1823884857344,32575705493504,612054254936064,

%U 12071987619713024,249477777420304384,5392386599983366144

%N Column 3 of triangle A111553.

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

%H Glenn Bruda, <a href="https://arxiv.org/abs/2412.19866">Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals</a>, arXiv:2412.19866 [math.CO], 2024. See page 4.

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

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

%F From _Groux Roland_, Dec 10 2010: (Start)

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

%F 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)

%F a(n) = Sum_{k=0..n} A200659(n,k)*3^k. - _Philippe Deléham_, Nov 21 2011

%F G.f.: 1/(1-4x/(1-2x/(1-5x/(1-3x/(1-6x/(1-4x/(1-...(continued fraction). - _Philippe Deléham_, Nov 21 2011

%F 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

%F 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

%F 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

%o (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])}

%Y Cf. A111553, A111531, A111559.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 07 2005