A192749 - OEIS

%I #11 Jun 13 2015 00:53:53

%S 0,1,6,16,35,68,124,217,370,620,1027,1688,2760,4497,7310,11864,19235,

%T 31164,50468,81705,132250,214036,346371,560496,906960,1467553,2374614,

%U 3842272,6216995,10059380,16276492,26335993,42612610,68948732,111561475

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

%C 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.

%C 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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).

%F 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

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

%t p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 n + 1;

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

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

%t FixedPoint[(s PolynomialQuotient @@ #1 +

%t        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}]

%t (* A053311 *)

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

%t (* A192749 *)

%Y Cf. A192744, A192232.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Jul 09 2011