A193005 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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

Author

Clark Kimberling, Jul 14 2011

Keywords

• nonn

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.

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Crossrefs

Cf. A192232, A192744, A192951, A193004, A000045.

Programs

Mathematica

(See A193004.)

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