login
A291220
p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S - S^4.
2
1, 1, 2, 4, 7, 15, 27, 55, 101, 199, 370, 718, 1347, 2595, 4898, 9397, 17803, 34066, 64682, 123561, 234917, 448289, 852979, 1626689, 3096695, 5903316, 11241426, 21424775, 40805833, 77759648, 148118585, 282229961, 537636210, 1024373916, 1951472023, 3718072991
OFFSET
0,3
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 A291219 for a guide to related sequences.
LINKS
FORMULA
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 5*a(n-4) + 3*a(n-5) + 4*a(n-6) - a(n-7) - a(n-8) for n >= 9.
G.f.: (1 + x - x^2)*(1 - x - x^2 + x^3 + x^4) / ((1 - x - 2*x^2 + x^4)*(1 - 2*x^2 + x^3 + x^4)). - Colin Barker, Aug 25 2017
MATHEMATICA
z = 60; s = x/(1 - x^2); p = 1 - s - s^4;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291220 *)
PROG
(PARI) Vec((1 + x - x^2)*(1 - x - x^2 + x^3 + x^4) / ((1 - x - 2*x^2 + x^4)*(1 - 2*x^2 + x^3 + x^4)) + O(x^30)) \\ Colin Barker, Aug 25 2017
CROSSREFS
Sequence in context: A232464 A264292 A259592 * A299099 A248574 A136336
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 24 2017
STATUS
approved