%I #27 Sep 18 2017 21:45:23
%S 0,1,4,9,18,33,58,99,166,275,452,739,1204,1957,3176,5149,8342,13509,
%T 21870,35399,57290,92711,150024,242759,392808,635593,1028428,1664049,
%U 2692506,4356585,7049122,11405739,18454894,29860667,48315596,78176299
%N Coefficient of x in 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)+n+2 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.
%C Form an array with m(1,j) = m(j,1) = j for j >= 1 in the top row and left column, and internal terms m(i,j) = m(i-1,j-1) + m(i-1,j). The sum of the terms in the n-th antidiagonal is a(n). - _J. M. Bergot_, Nov 07 2012
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F a(n) = 2*A000045(n+3)-n-4. G.f. x*(-1-x+x^2) / ( (x^2+x-1)*(x-1)^2 ). - _R. J. Mathar_, Nov 09 2012
%F a(n) = Sum_{1..n} C(n-i+2,i+1) + C(n-i,i). - _Wesley Ivan Hurt_, Sep 13 2017
%t q = x^2; s = x + 1; z = 40;
%t p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 2;
%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}] (* A001594 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192760 *)
%Y Cf. A192744, A192232.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Jul 09 2011