%I #4 Sep 11 2017 19:48:37
%S 0,0,1,0,3,2,3,12,4,30,27,45,108,90,260,342,498,1115,1218,2709,3913,
%T 5949,11469,15262,28461,44556,68028,123243,178650,311337,498114,
%U 777996,1340603,2052765,3435906,5569902,8800392,14783823,23242761,38249550,62156709
%N p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S^3 - S^6.
%C Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
%C See A291728 for a guide to related sequences.
%H Clark Kimberling, <a href="/A291739/b291739.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 1, 0, 3, 1, 3, 6, 1, 15, 0, 20, 0, 15, 0, 6, 0, 1)
%F G.f.: -((x^2 (1 + x^2)^3 (1 + x + x^2) (1 + x + x^3) (1 - 2 x + 2 x^2 - x^3 + x^4))/(-1 + x^3 + 3 x^5 + x^6 + 3 x^7 + 6 x^8 + x^9 + 15 x^10 + 20 x^12 + 15 x^14 + 6 x^16 + x^18)).
%F a(n) = a(n-3) + 3*a(n-5) + a(n-6) + 3*a(n-7) + 6*a(n-8) + a(n-9) + 15*a(n-10) + 20 *a(n-12) + 15*a(n-14) + 6*a(n-16) + a(n-18) for n >= 19.
%t z = 60; s = x + x^3; p = 1 - s^3 - s^6;
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A154272 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291739 *)
%Y Cf. A154272, A291728.
%K nonn,easy
%O 0,5
%A _Clark Kimberling_, Sep 11 2017