cp's OEIS Frontend

A291738 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^4.

%I A291738 #4 Sep 11 2017 20:05:29
%S A291738 1,1,2,4,6,13,23,43,76,138,244,444,795,1444,2600,4705,8474,15307,
%T A291738 27583,49797,89800,162088,292388,527663,951922,1717692,3098937,
%U A291738 5591589,10088361,18202665,32841990,59256835,106914493,192904396,348050363,627980316,1133045001
%N A291738 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^4.
%C A291738 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 A291738 See A291728 for a guide to related sequences.
%H A291738 Clark Kimberling, <a href="/A291738/b291738.txt">Table of n, a(n) for n = 0..1000</a>
%H A291738 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1)
%F A291738 G.f.: -(((1 + x^2) (1 + x + x^2) (1 + x + x^3) (1 - 2 x + 2 x^2 - x^3 + x^4))/(-1 + x + x^3 + x^4 + 4 x^6 + 6 x^8 + 4 x^10 + x^12)).
%F A291738 a(n) = a(n-1) + a(n-3) + a(n-4) + 4* a(n-6) + 6*a(n-8) + 4*a(n-10) + a(n-12) for n >= 13.
%t A291738 z = 60; s = x + x^3; p = 1 - s - s^4;
%t A291738 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)
%t A291738 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291738 *)
%Y A291738 Cf. A154272, A291728.
%K A291738 nonn,easy
%O A291738 0,3
%A A291738 _Clark Kimberling_, Sep 11 2017