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A193044
Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
2
1, 0, 2, 5, 13, 28, 56, 105, 189, 330, 564, 949, 1579, 2606, 4276, 6987, 11383, 18506, 30042, 48719, 78951, 127880, 207062, 335195, 542533, 878028, 1420886, 2299265, 3720529, 6020200, 9741164, 15761829, 25503489, 41265846, 66769896
OFFSET
0,3
COMMENTS
The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n(-1+n^2)/6, 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.: ( 1+7*x^2-4*x^3+x^4-4*x ) / ( (x^2+x-1)*(x-1)^3 ). - R. J. Mathar, May 04 2014
MATHEMATICA
q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n (n^2 - 1)/6;
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}]
(* A193044 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A193045 *)
CROSSREFS
Cf. A192232, A192744, A192951, A193045, A179991 (first differences).
Sequence in context: A026522 A216378 A225690 * A122491 A320933 A290194
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 15 2011
STATUS
approved