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A192745
Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
2
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.
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 09 2011
STATUS
approved