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

0,3

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.

Links

Table of n, a(n) for n=0..33.

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

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

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

Sequence in context: A361493 A274326 A055329 * A152895 A308524 A335629

Adjacent sequences: A193002 A193003 A193004 * A193006 A193007 A193008

Keyword

nonn

Author

Clark Kimberling, Jul 14 2011

Status

approved