A193044 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
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
Links
- Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
Programs
-
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 *)
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
Comments