A192745 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

0, 1, 2, 5, 13, 42, 175, 937, 6152, 47409, 416441, 4092650, 44425891, 527520141, 6798966832, 94504778173, 1408978113005, 22426272779178, 379522678988183, 6804322657495361, 128828945745315544, 2568535276579450905, 53788306394034206449

Offset: 0

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn

Comments

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+n! for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

Examples

			The first six polynomials and their reductions are shown here:
1 -> 1
1+x -> 1+x
2+x+x^2 -> 3+2x
6+2x+x^2+x^3 -> 8+5x
24+6x+2x^2+x^4+x^5 -> 29+13x
From those, read A192744=(1,1,3,8,29,...) and A192745=(0,1,2,5,13,...).
		

Crossrefs

A192744, A192232.

Programs

Mathematica

(See A192744.)

Formula

G.f.: x/(1-x-x^2)/Q(0), where Q(k)= 1 - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013

Conjecture: a(n) -n*a(n-1) +(n-2)*a(n-2) +(n-1)*a(n-3)=0. - R. J. Mathar, May 04 2014

a(n) = Sum_{k=0..n} k!*Fibonacci(n-k). - Greg Dresden, Dec 03 2021

a(n) ~ (n-1)!. - Vaclav Kotesovec, Dec 03 2021