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A193048
Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
2
1, 0, 1, 2, 8, 25, 68, 163, 357, 730, 1417, 2642, 4774, 8417, 14556, 24793, 41729, 69582, 115187, 189614, 310786, 507715, 827356, 1345697, 2185703, 3546350, 5749603, 9316428, 15089782, 24433615, 39554862, 64024437, 103620219, 167691032
OFFSET
0,4
COMMENTS
The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n(4-5*n^2+n^4)/120, 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) = 6*a(n-1)-14*a(n-2)+15*a(n-3)-5*a(n-4)-4*a(n-5)+4*a(n-6)-a(n-7).
G.f.: (x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1) / ((x-1)^5*(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 (-1 + n^2) (-4 + n^2)/120;
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}] (* A193048 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193049 *)
PROG
(PARI) Vec((x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1)/((x-1)^5*(x^2+x-1)) + O(x^100)) \\ Colin Barker, May 12 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 15 2011
STATUS
approved