OFFSET
0,3
COMMENTS
if s(n) is a sequence defined as s(0)=x, s(n) = 2n*s(n-1)+k, n>0, then s(n) = 2^n*n!*x + a(n)*k. - Gary Detlefs, Feb 20 2010
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..403
FORMULA
a(n) = Sum[P(n, k) * 2^k {k=0 to n-1}] - Ross La Haye, Sep 15 2004
Conjecture: a(n) +(-2*n-1)*a(n-1) +2*(n-1)*a(n-2)=0. - R. J. Mathar, May 29 2013
E.g.f.: (exp(x)-1)/(1-2*x) = -12*x/(Q(0)+6*x-3*x^2)/(1-2*x), where Q(k) = 2*(4*k+1)*(32*k^2+16*k+x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
EXAMPLE
a(3) = 2*3*a(2)+1 = 6*5+1 = 31.
MATHEMATICA
nxt[{n_, a_}]:={n+1, 2a(n+1)+1}; NestList[nxt, {0, 0}, 20][[All, 2]] (* or *) With[{nn=20}, CoefficientList[Series[(Exp[x]-1)/(1-2x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Aug 08 2021 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Henry Bottomley, Jun 20 2000
EXTENSIONS
More terms from James A. Sellers, Jul 04 2000
STATUS
approved