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

1, 1, 7, 17, 47, 110, 250, 538, 1123, 2278, 4522, 8812, 16911, 32031, 59991, 111263, 204593, 373370, 676800, 1219440, 2185251, 3896796, 6917892, 12231192, 21544717, 37819885, 66179335, 115464893, 200906723, 348688838

Offset: 1

Author

Clark Kimberling, Jun 26 2011

Keywords

• nonn
• changed

Comments

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

Crossrefs

Cf. A192232, A192245.

Programs

Mathematica

c[n_] := n (n + 1)/2;  (* triangular numbers, A000217 *)
Table[c[n], {n, 1, 15}]
q[x_] := x + 1; p[0, x_] := 1;
p[n_, x_] :=  p[n - 1, x] + (x^n)*c[n]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t,n],x,0],{n,1,30}] (* A192244 *)
Table[Coefficient[Part[t,n],x,1],{n,1,30}] (* A192245 *)
(* Peter J. C. Moses, Jun 26 2011 *)

Formula

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