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A192761 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192761 #22 Feb 19 2025 02:40:54
%S A192761 0,1,5,11,22,40,70,119,199,329,540,882,1436,2333,3785,6135,9938,16092,
%T A192761 26050,42163,68235,110421,178680,289126,467832,756985,1224845,1981859,
%U A192761 3206734,5188624,8395390,13584047,21979471,35563553,57543060,93106650
%N A192761 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192761 The titular polynomial is defined recursively by p(n,x) = x*(n-1,x) + n + 3 for n > 0, where p(0,x) = 1.  For discussions of polynomial reduction, see A192232 and A192744.
%C A192761 Construct a triangle with T(n,0) = n*(n+1)+1 and T(n,n) = (n+1)*(n+2)/2 starting at n=0. Define the interior terms by T(r,c) = T(r-2,c-1) + T(r-1,c). The sequence of its row sums is 1, 6, 17, 39, 79, 149, 268, 467,... and the first differences of these (the sum of the terms in row(n) less those in row(n-1)) equals a(n+1). - _J. M. Bergot_, Mar 10 2013
%H A192761 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192761 a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(2*x^2-2*x-1) / ((x-1)^2*(x^2+x-1)). [_Colin Barker_, Dec 08 2012]
%t A192761 q = x^2; s = x + 1; z = 40;
%t A192761 p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 3;
%t A192761 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192761 reduce[{p1_, q_, s_, x_}] :=
%t A192761 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192761        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192761 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192761 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192761   (* A022318 *)
%t A192761 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192761   (* A192761 *)
%Y A192761 Cf. A192744, A192232, partial sums of A022318.
%K A192761 nonn,easy
%O A192761 0,3
%A A192761 _Clark Kimberling_, Jul 09 2011