A056541 a(n) = 2n*a(n-1) + 1 with a(0)=0.

0, 1, 5, 31, 249, 2491, 29893, 418503, 6696049, 120528883, 2410577661, 53032708543, 1272785005033, 33092410130859, 926587483664053, 27797624509921591, 889523984317490913, 30243815466794691043

Offset: 0

Author

Henry Bottomley, Jun 20 2000

Keywords

• easy
• nonn

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

Examples

			a(3) = 2*3*a(2)+1 = 6*5+1 = 31.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..403

Crossrefs

Cf. A002627, A173516.

Programs

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 *)

Formula

a(n) = floor[(sqrt(e)-1)*2^n*n! ] = A010844(n)-A000165(n).

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

Extensions

More terms from James Sellers, Jul 04 2000