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A291394
p-INVERT of (1,1,0,0,0,0,...), where p(S) = (1 - S)(1 - 3 S).
2
4, 17, 66, 254, 968, 3679, 13962, 52957, 200812, 761396, 2886768, 10944725, 41494856, 157319353, 596443614, 2261290498, 8573204920, 32503490435, 123230092830, 467200760741, 1771292578424, 6715480046152, 25460317920096, 96527393973769, 365963135802988
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 A291382 for a guide to related sequences.
FORMULA
G.f.: -(((1 + x) (-4 + 3 x + 3 x^2))/((-1 + x + x^2) (-1 + 3 x + 3 x^2))).
a(n) = 4*a(n-1) + a(n-2) - 6*a(n-3) - 3*a(n-4) for n >= 5.
MATHEMATICA
z = 60; s = x + x^2; p = (1 - s)(1 - 3s);
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A019590 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291394 *)
CROSSREFS
Sequence in context: A045992 A217539 A046723 * A244616 A030529 A266862
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 06 2017
STATUS
approved