|
|
A291038
|
|
p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = (1 - 2 S)^2.
|
|
2
|
|
|
4, 12, 32, 84, 216, 544, 1348, 3300, 8000, 19236, 45936, 109056, 257604, 605820, 1419232, 3313396, 7711944, 17900320, 41445764, 95746260, 220735616, 507934276, 1166792160, 2676017408, 6128381316, 14015556588, 32012831648, 73033858964, 166434905016
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: -((4 (-1 + x + x^3))/(-1 + 2 x + x^3)^2).
a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3) - 4*a(n-4) - a(n-6) for n >= 7.
|
|
MATHEMATICA
|
z = 60; s = x/(x - x^3); p = (1 - 2 s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A079978 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291038 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|