%I #8 Dec 04 2016 19:46:25

%S 1,1,13,35,105,258,608,1344,2865,5910,11894,23444,45427,86755,163645,

%T 305397,564647,1035446,1885050,3409610,6131441,10968416,19528188,

%U 34617960,61125685,107540053,188567053,329625719,574558965,998836650

%N 0-sequence of reduction of pentagonal numbers sequence by x^2 -> x+1.

%C See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

%F Empirical G.f.: x*(1-3*x+12*x^2-9*x^3+4*x^4)/(1-x)/(1-x-x^2)^3. [Colin Barker, Feb 11 2012]

%t Remove["Global`*"];

%t c[n_] := n (3 n - 1)/2; (* pentagonal numbers, A000326 *)

%t Table[c[n], {n, 1, 15}]

%t q[x_] := x + 1;

%t p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]

%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),

%t x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[

%t Last[Most[

%t FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,

%t 30}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192145 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192146 *)

%t (* by _Peter J. C. Moses_, Jun 20 2011 *)

%Y Cf. A192232, A192146.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jun 27 2011