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A193004
Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
2
1, 1, 9, 29, 75, 165, 331, 623, 1123, 1963, 3357, 5651, 9405, 15525, 25477, 41633, 67831, 110281, 179031, 290339, 470511, 762111, 1234009, 1997639, 3233305, 5232745, 8468001, 13702853, 22173123, 35878413, 58054147, 93935351, 151992475
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) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).
G.f.: (x^4-3*x^3+10*x^2-3*x+1) / ((x-1)^3*(x^2+x-1)). - Colin Barker, May 12 2014
MATHEMATICA
q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^3;
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}] (* A193004 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193005 *)
PROG
(PARI) Vec((x^4-3*x^3+10*x^2-3*x+1)/((x-1)^3*(x^2+x-1)) + O(x^100)) \\ Colin Barker, May 12 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 14 2011
STATUS
approved