A192745 - OEIS

%I #15 Dec 03 2021 09:27:23

%S 0,1,2,5,13,42,175,937,6152,47409,416441,4092650,44425891,527520141,

%T 6798966832,94504778173,1408978113005,22426272779178,379522678988183,

%U 6804322657495361,128828945745315544,2568535276579450905,53788306394034206449

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

%C 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.

%F 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

%F Conjecture: a(n) -n*a(n-1) +(n-2)*a(n-2) +(n-1)*a(n-3)=0. - _R. J. Mathar_, May 04 2014

%F a(n) = Sum_{k=0..n} k!*Fibonacci(n-k). - _Greg Dresden_, Dec 03 2021

%F a(n) ~ (n-1)!. - _Vaclav Kotesovec_, Dec 03 2021

%e The first six polynomials and their reductions are shown here:

%e 1 -> 1

%e 1+x -> 1+x

%e 2+x+x^2 -> 3+2x

%e 6+2x+x^2+x^3 -> 8+5x

%e 24+6x+2x^2+x^4+x^5 -> 29+13x

%e From those, read A192744=(1,1,3,8,29,...) and A192745=(0,1,2,5,13,...).

%t (See A192744.)

%Y A192744, A192232.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jul 09 2011