OFFSET
1,1
COMMENTS
See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
FORMULA
a(n) = 2*A190062(n).
G.f.: 2*x*(1-2*x+2*x^2)/((1-x)*(1-x-x^2)^2). [Colin Barker, Feb 11 2012]
MATHEMATICA
c[n_] := 2 n; (* even numbers, A005843 *)
Table[c[n], {n, 1, 15}]
q[x_] := x + 1;
p[0, x_] := 2; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
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,
40}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}] (* A192305 *)
Table[Coefficient[Part[t, n]/2, x, 0], {n, 1, 40}] (* A190062 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}] (* A192306 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}] (* A122491 *)
(* by Peter J. C. Moses, Jun 20 2011 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 27 2011
STATUS
approved