A290930 - OEIS

%I #6 Aug 19 2017 13:24:40

%S 0,3,12,37,116,372,1188,3763,11860,37261,116760,365056,1139224,

%T 3549635,11045804,34335421,106633804,330916268,1026277180,3181108619,

%U 9855901108,30524529485,94506627952,292521594048,905220237168,2800700318291,8663793207244

%N p-INVERT of the positive integers, where p(S) = (1 - S^2)(1 - 2*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 Clark Kimberling, <a href="/A290930/b290930.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8, -25, 44, -54, 44, -25, 8, -1)

%F G.f.: (3 x - 12 x^2 + 16 x^3 - 12 x^4 + 3 x^5)/(1 - 8 x + 25 x^2 - 44 x^3 +  54 x^4 - 44 x^5 + 25 x^6 - 8 x^7 + x^8).

%F a(n) = 8*a(n-1) - 25*a(n-2) + 44*a(n-3) - 54*a(n-4) + 44*a(n-5) - 25*a(n-6) + 8*a(n-7) - a(n-8).

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

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

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290930 *)

%Y Cf. A000027, A290890.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Aug 19 2017