A192749 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

0, 1, 6, 16, 35, 68, 124, 217, 370, 620, 1027, 1688, 2760, 4497, 7310, 11864, 19235, 31164, 50468, 81705, 132250, 214036, 346371, 560496, 906960, 1467553, 2374614, 3842272, 6216995, 10059380, 16276492, 26335993, 42612610, 68948732, 111561475

Offset: 0

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn
• easy

Comments

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+4n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

a(n+1) is the row sum of row n of the triangle defined by T(n,1)=n*(n-1)+1, T(n,n)=2*n-1, n>=1, and T(r,c)=T(r-1,c)+T(r-2,c-1). The triangle starts 1; 3,3; 7,4,5; 13,7,8,7; 21,14,12,12,9; - J. M. Bergot, Apr 26 2013

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Cf. A192744, A192232.

Programs

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 n + 1;
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}]
(* A053311 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192749 *)

Formula

G.f. -x*(1+3*x) / ( (x^2+x-1)*(x-1)^2 ). a(n+1)-a(n) = A053311(n). - R. J. Mathar, Apr 29 2013