A111546 Column 2 of triangle A111544.

1, 3, 15, 99, 783, 7083, 71415, 789939, 9485343, 122721723, 1701224775, 25156450179, 395362560303, 6583219735563, 115817825451735, 2147443419579219, 41868118883289663, 856527397513863003

Offset

0,2

Comments

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

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

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

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]

a(n) = Sum_{k=0..n} A200659(n,k)*2^k. - Philippe Deléham, Nov 21 2011

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

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

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

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

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

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

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

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

Prog

(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])}
(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
(Haskell)
a111546 n = a111546_list !! n
a111546_list = 1 : f 2 [1] where
   f v ws@(w:_) = y : f (v + 1) (y : ws) where
                  y = v * w + (sum $ zipWith (*) ws $ reverse ws)
-- Reinhard Zumkeller, Jan 24 2014

Crossrefs

Cf. A111544, A111530, A111548, A230253.

Cf. A005412.

Sequence in context: A168344 A091713 A156106 * A219359 A152402 A361596

Adjacent sequences: A111543 A111544 A111545 * A111547 A111548 A111549

Keyword

nonn,changed

Author

Paul D. Hanna, Aug 07 2005

Status

approved