A193004 - OEIS

%I #12 Jun 13 2015 00:53:55

%S 1,1,9,29,75,165,331,623,1123,1963,3357,5651,9405,15525,25477,41633,

%T 67831,110281,179031,290339,470511,762111,1234009,1997639,3233305,

%U 5232745,8468001,13702853,22173123,35878413,58054147,93935351,151992475

%N Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

%C 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.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).

%F a(n) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).

%F 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

%t q = x^2; s = x + 1; z = 40;

%t p[0, x] := 1;

%t p[n_, x_] := x*p[n - 1, x] + n^3;

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A193004 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193005 *)

%o (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

%Y Cf. A192232, A192744, A192951, A193005.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Jul 14 2011