A290906 - OEIS

%I #8 Aug 18 2017 18:46:22

%S 0,3,12,39,132,456,1572,5409,18612,64053,220440,758640,2610840,

%T 8985147,30922188,106418031,366235308,1260390744,4337606988,

%U 14927778921,51373622388,176801189997,608457401520,2093992746720,7206429919920,24800769855603,85351303248012

%N p-INVERT of the positive integers, where p(S) = 1 - 3*S^2.

%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 A290890 for a guide to related sequences.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -3, 4, -1)

%F G.f.: (3 x)/(1 - 4 x + 3 x^2 - 4 x^3 + x^4).

%F a(n) = 4*a(n-1) - 3*a(n-2) + 4*a(n-3) - a(n-4).

%F a(n) = 3*A290907(n) for n >= 0.

%t z = 60; s = x/(1 - x)^2; p = 1 - 3 s^2;

%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)

%t u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290906 *)

%t u/3 (* A290907 *)

%Y Cf. A000027, A290890, A290907.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Aug 17 2017