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A291728
p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2.
31
1, 2, 4, 9, 17, 35, 70, 142, 285, 576, 1160, 2340, 4716, 9510, 19171, 38653, 77926, 157110, 316747, 638599, 1287479, 2595698, 5233196, 10550681, 21271280, 42885152, 86460984, 174314476, 351436368, 708532813, 1428476905, 2879960190, 5806303628, 11706120825
OFFSET
0,2
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).
In the following guide to p-INVERT sequences using s = (1,0,1,0,0,0,0,...) = A154272, in some cases t(1,0,1,0,0,0,0,...) is a shifted (or differently indexed) version of the indicated sequence:
***
p(S) t(1,0,1,0,0,0,0,...)
1 - S A000930 (Narayana's cows sequence)
1 - S^2 A002478 (except for 0's)
1 - S^3 A291723
1 - S^5 A291724
(1 - S)^2 A291725
(1 - S)^3 A291726
(1 - S)^4 A291727
1 - S - S^2 A291728
1 - 2S - S^2 A291729
1 - 2S - 2S^2 A291730
(1 - 2S)^2 A291732
(1 - S)(1 - 2S) A291734
1 - S - S^3 A291735
1 - S^2 - S^3 A291736
1 - S - S^2 - S^3 A291737
1 - S - S^4 A291738
1 - S^3 - S^6 A291739
(1 - S)(1 - S^2) A291740
(1 - S)(1 + S^2) A291741
FORMULA
G.f.: (-1 - x - x^2 - 2 x^3 - x^5)/(-1 + x + x^2 + x^3 + 2 x^4 + x^6).
a(n) = a(n-1) + a(n-2) + a(n-3) + 2*(a(n-4) + a(n-6) for n >= 7.
MATHEMATICA
z = 60; s = x + x^3; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A154272 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291728 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 08 2017
STATUS
approved