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

0,3

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.

Links

Table of n, a(n) for n=0..22.

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

Example

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,...).

Mathematica

(See A192744.)

Crossrefs

A192744, A192232.

Sequence in context: A114297 A178682 A229161 * A299430 A307263 A278171

Adjacent sequences: A192742 A192743 A192744 * A192746 A192747 A192748

Keyword

nonn

Author

Clark Kimberling, Jul 09 2011

Status

approved