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A192244 0-sequence of reduction of triangular number sequence by x^2 -> x+1.

%I A192244 #15 Aug 23 2025 08:03:53
%S A192244 1,1,7,17,47,110,250,538,1123,2278,4522,8812,16911,32031,59991,111263,
%T A192244 204593,373370,676800,1219440,2185251,3896796,6917892,12231192,
%U A192244 21544717,37819885,66179335,115464893,200906723,348688838
%N A192244 0-sequence of reduction of triangular number sequence by x^2 -> x+1.
%C A192244 See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%F A192244 Empirical g.f.: x*(1-3*x+6*x^2-3*x^3)/(1-x)/(1-x-x^2)^3. - _Colin Barker_, Feb 10 2012
%t A192244 c[n_] := n (n + 1)/2;  (* triangular numbers, A000217 *)
%t A192244 Table[c[n], {n, 1, 15}]
%t A192244 q[x_] := x + 1; p[0, x_] := 1;
%t A192244 p[n_, x_] :=  p[n - 1, x] + (x^n)*c[n]
%t A192244 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192244    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192244 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
%t A192244 Table[Coefficient[Part[t,n],x,0],{n,1,30}] (* A192244 *)
%t A192244 Table[Coefficient[Part[t,n],x,1],{n,1,30}] (* A192245 *)
%t A192244 (* _Peter J. C. Moses_, Jun 26 2011 *)
%Y A192244 Cf. A192232, A192245.
%K A192244 nonn,changed
%O A192244 1,3
%A A192244 _Clark Kimberling_, Jun 26 2011