A098557 Expansion of e.g.f. (1/2)*(1+x)*log((1+x)/(1-x)).

0, 1, 2, 2, 8, 24, 144, 720, 5760, 40320, 403200, 3628800, 43545600, 479001600, 6706022400, 87178291200, 1394852659200, 20922789888000, 376610217984000, 6402373705728000, 128047474114560000, 2432902008176640000, 53523844179886080000, 1124000727777607680000

Offset: 0

Author

Paul Barry, Sep 14 2004

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..450

Crossrefs

From Johannes W. Meijer, Nov 12 2009: (Start)

Cf. A109613 (odd numbers repeated).

Equals the first left hand column of A167552.

Equals the first right hand column of A167556.

A098557(n)*A064455(n) equals the second right hand column of A167556(n).

(End)

Cf. A052558, A073742.

Programs

Magma

[0,1] cat [Factorial(n-1) + Factorial(n-2)*(1+(-1)^n)/2: n in [2..30]]; // G. C. Greubel, Jan 17 2018

Mathematica

Join[{0,1}, Table[(n-1)! + (n-2)!*(1+(-1)^n)/2, {n,2,30}]] (* or *) With[{nmax = 50}, CoefficientList[Series[(1/2)*(1 + x)*Log[(1 + x)/(1 - x)], {x,0,nmax}], x]*Range[0,nmax]!] (* G. C. Greubel, Jan 17 2018 *)

PARI

for(n=0, 30, print1(if(n==0,0, if(n==1, 1, (n-1)! + (n-2)!*(1 + (-1)^n)/2)), ", ")) \\ G. C. Greubel, Jan 17 2018

Formula

a(n+1) = n! + (n-1)! * (1-(-1)^n)/2.

a(n+2) = 2*A052558(n).

conjecture: -a(n) +a(n-1) +(n-1)*(n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011

G.f.: 1-G(0), where G(k)= 1 + x*(2*k-1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013

Sum_{n>=1} 1/a(n) = sinh(1) + 1 = A073742 + 1. - Amiram Eldar, Jan 22 2023