A292402 - OEIS

A292402

p-INVERT of (1,0,0,1,0,0,0,0,0,0,...), where p(S) = 1 - S^2.

0, 1, 0, 1, 2, 1, 4, 2, 6, 7, 8, 16, 14, 29, 32, 47, 70, 82, 136, 162, 244, 331, 440, 650, 834, 1220, 1632, 2262, 3176, 4261, 6056, 8175, 11414, 15747, 21568, 30121, 41094, 57210, 78644, 108521, 150300, 206456, 286288, 393865, 544424, 751675, 1035980

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OFFSET

0,5

COMMENTS

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).

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 0, 2, 0, 0, 1)

FORMULA

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

a(n) = a(n-2) + 2*a(n-5) + a(n-8) for n >= 9.

MATHEMATICA

z = 60; s = x + x^4; p = 1 - s^2;

Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]

Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292402 *)

CROSSREFS

Cf. A292324, A292403, A292404.