A111546 - OEIS

%I #50 Jan 06 2025 06:29:58

%S 1,3,15,99,783,7083,71415,789939,9485343,122721723,1701224775,

%T 25156450179,395362560303,6583219735563,115817825451735,

%U 2147443419579219,41868118883289663,856527397513863003

%N Column 2 of triangle A111544.

%C Also forms the columns of triangle A111548, which is the matrix inverse of triangle A111544.

%H Reinhard Zumkeller, <a href="/A111546/b111546.txt">Table of n, a(n) for n = 0..250</a>

%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+2)!/2!*x^n) = Sum_{n>=1} a(n)*x^n/n. a(n) = 3*A111530(n) = -A111548(n+1, 0) for n>0.

%F a(n+1) = (1/2)*((n+4)!-3*(n+3)!-Sum_{k=0..n-1} (n+2-k)!*a(k+1).

%F a(n+1) is the moment of order n for the measure of density: 2*x^2*exp(-x)/((x^2*exp(-x)*Ei(x)-x-1)^2+Pi^2*x^4*exp(-2*x)), on the interval 0..infinity. [_Groux Roland_, Dec 10 2010]

%F a(n) = Sum_{k=0..n} A200659(n,k)*2^k. - _Philippe Deléham_, Nov 21 2011

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

%F G.f. 2 - U(0) where U(k)= 1 - x*(k+1)/(1 - x*(k+3)/U(k+1)); (continued fraction, 2-step). - _Sergei N. Gladkovskii_, Jun 29 2012

%F G.f. -1/G(0) where G(k) = x - 1 - k*x - x*(k+2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - _Sergei N. Gladkovskii_, Aug 13 2012

%F G.f.: A(x) = 1/(G(0) - x) where G(k) = 1 + (k+1)*x - x*(k+3)/G(k+1) ; (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 26 2012

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

%F G.f.: 1/x -2 -2/(x*G(0)), where G(k)= 1 + 1/(1 - x*(k+3)/(x*(k+3) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 06 2013

%F G.f.: 1/x - 2 - 1/(x*W(0)), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Aug 25 2013

%F G.f.: W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 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+3,n+3,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+2)!/2!*x^i)),m-j-1))))^-1)[n+3,3])}

%o (PARI) a(n)=(1/2)*((n+3)!-3*(n+2)!-sum(k=0,n-2,(n+1-k)!*a(k+1))) \\ Formula by R. Groux, implemented & checked to conform to given terms by _M. F. Hasler_, Dec 12 2010

%o (Haskell)

%o a111546 n = a111546_list !! n

%o a111546_list = 1 : f 2 [1] where

%o    f v ws@(w:_) = y : f (v + 1) (y : ws) where

%o                   y = v * w + (sum $ zipWith (*) ws $ reverse ws)

%o -- _Reinhard Zumkeller_, Jan 24 2014

%Y Cf. A111544, A111530, A111548, A230253.

%Y Cf. A005412.

%K nonn,changed

%O 0,2

%A _Paul D. Hanna_, Aug 07 2005