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A291411 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2 + S^3. 2
2, 7, 21, 63, 189, 567, 1699, 5092, 15260, 45731, 137046, 410697, 1230768, 3688339, 11053134, 33123790, 99264648, 297474121, 891463923, 2671519536, 8005951162, 23992058879, 71898875923, 215464974683, 645700711159, 1935021731510, 5798830691535, 17377808652745 (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 A291382 for a guide to related sequences.
LINKS
FORMULA
G.f.: -(((-1 + x) (1 + x) (2 + x) (1 + x + x^2))/(1 - 2 x - 3 x^2 - x^3 + 2 x^4 + 3 x^5 + x^6)).
a(n) = 2*a(n-1) + 3*a(n-2) + a(n-3) - 2*a(n-4) - 3*a(n-5) - a(n-6) for n >= 7.
MATHEMATICA
z = 60; s = x + x^2; p = 1 - 2 s - s^2 + s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A019590 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291411 *)
PROG
(GAP)
a:=[2, 7, 21, 63, 189, 567];; for n in [7..10^2] do a[n]:=2*a[n-1]+3*a[n-2]+a[n-3]-2*a[n-4]-3*a[n-5]-a[n-6]; od; a; # Muniru A Asiru, Sep 12 2017
CROSSREFS
Sequence in context: A320811 A005666 A353094 * A159972 A106271 A027990
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 07 2017
STATUS
approved

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Last modified June 24 09:22 EDT 2024. Contains 373674 sequences. (Running on oeis4.)