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A291729
p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2.
2
2, 5, 14, 39, 106, 290, 794, 2173, 5946, 16272, 44530, 121860, 333480, 912597, 2497400, 6834349, 18702782, 51181767, 140063294, 383295214, 1048920220, 2870460125, 7855260268, 21496593296, 58827270844, 160985870984, 440551640160, 1205607339709, 3299247863502
OFFSET
0,1
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).
See A291728 for a guide to related sequences.
FORMULA
G.f.: (-2 - x - 2 x^2 - 2 x^3 - x^5)/(-1 + 2 x + x^2 + 2 x^3 + 2 x^4 + x^6).
a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) + 2*a(n-4) + a(n-6) for n >= 7.
MATHEMATICA
z = 60; s = x + x^3; p = 1 - 2 s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A154272 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291729 *)
CROSSREFS
Sequence in context: A148315 A331573 A141752 * A142586 A202207 A132834
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 11 2017
STATUS
approved