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A193046
Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
3
1, 1, 17, 83, 275, 727, 1673, 3505, 6873, 12843, 23155, 40639, 69889, 118353, 198097, 328659, 541667, 888311, 1451433, 2365089, 3846201, 6245771, 10131747, 16423103, 26606785, 43088737, 69761873, 112925075, 182770163, 295787863
OFFSET
0,3
COMMENTS
The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n^4, 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^5-6*x^4-x^3-21*x^2+4*x-1) / ((x-1)^4*(x^2+x-1)). - Colin Barker, May 11 2014
MATHEMATICA
q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^4;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A193046 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193047 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 15 2011
STATUS
approved