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A193005
Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
1
0, 1, 2, 11, 40, 115, 280, 611, 1234, 2357, 4320, 7677, 13328, 22733, 38258, 63735, 105368, 173199, 283480, 462511, 752850, 1223361, 1985472, 3219481, 5217120, 8450425, 13683170, 22151171, 35854024, 58027147, 93905560, 151959707, 245895058, 397887533
OFFSET
0,3
COMMENTS
The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.
FORMULA
a(n) = 5*a(n-1)-9*a(n-2)+6*a(n-3)+a(n-4)-3*a(n-5)+a(n-6).
G.f.: -x*(1-3*x+10*x^2-3*x^3+x^4) / ( (x^2+x-1)*(x-1)^4 ). - R. J. Mathar, May 12 2014
a(n) = 10*F(n+4) + 4*F(n+5) - 50 - 24*n - 6*n^2 - n^3, where F = A000045. - Greg Dresden, Jan 01 2021
MATHEMATICA
(See A193004.)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 14 2011
STATUS
approved