OFFSET
1,1
LINKS
Robert Israel, Table of n, a(n) for n = 1..449
FORMULA
a(n) = n!*Sum_{j=1..n} (-1)^(j+1)*(j+1)/j.
E.g.f.: (x+(x+1)*log(1+x))/(1-x^2). - Vladeta Jovovic, Mar 03 2003
Conjecture: -(n+1)*a(n+1) + a(n) + n^2*(n+2)*a(n-1) = 0. - R. J. Mathar, Sep 27 2012, corrected for offset 1 by Robert Israel, Dec 26 2018
Conjecture verified, using the differential equation (x^3-x)*g''(x) + (5*x^2-1)*g'(x) + (3*x+1)*g(x) + 2 = 0 satisfied by the e.g.f. - Robert Israel, Dec 26 2018
a(n) ~ n! * (log(2) + 1/2 - 1/2*(-1)^n). - Vaclav Kotesovec, Sep 29 2013
a(n) = n!*(log(2) + (n mod 2) - (-1)^n*LerchPhi(-1, 1, n+1)). - Peter Luschny, Dec 26 2018
MAPLE
f:= gfun:-rectoproc({-(n+1)*a(n+1) + a(n) + n^2*(n+2)*a(n-1)=0, a(1)=2, a(2)=1}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Dec 26 2018
MATHEMATICA
Rest[CoefficientList[Series[(x+(x+1)*Log[1+x])/(1-x^2), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
a[n_] := n!(Log[2] + Boole[OddQ[n]] - (-1)^n LerchPhi[-1, 1, 1 + n]);
Table[a[n], {n, 1, 20}] (* Peter Luschny, Dec 26 2018 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 01 2003
STATUS
approved