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A192250 0-sequence of reduction of central binomial coefficient sequence by x^2 -> x+1.

%I A192250 #8 Dec 04 2016 19:46:25
%S A192250 1,1,7,27,167,923,5543,32999,200309,1221329,7503033,46301793,
%T A192250 286971677,1784658077,11131825877,69611130917,436270168817,
%U A192250 2739539507957,17232530582057,108564692241257,684901029237677,4326215549824277,27357682806703397
%N A192250 0-sequence of reduction of central binomial coefficient sequence by x^2 -> x+1.
%C A192250 See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%F A192250 Conjecture: (n-1)*(n-2)*a(n) -(5*n-7)*(n-2)*a(n-1) -2*(2*n-3)*(3*n-8)*a(n-2) +4*(2*n-3)*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, May 04 2014
%t A192250 c[n_] := (2 n)!/(n! n!); (* central binomial coefficients, A000984 *)
%t A192250 Table[c[n], {n, 0, 15}]
%t A192250 q[x_] := x + 1;
%t A192250 p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n]
%t A192250 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192250    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192250 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
%t A192250    30}]
%t A192250 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192250 *)
%t A192250 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192251 *)
%t A192250 Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]  (* A192070 *)
%t A192250 (* by _Peter J. C. Moses_, Jun 20 2011 *)
%Y A192250 Cf. A192232, A192251, A192070.
%K A192250 nonn
%O A192250 1,3
%A A192250 _Clark Kimberling_, Jun 27 2011