A292403 p-INVERT of (1,0,0,0,0,1,0,0,0,0,0,0,...), where p(S) = 1 - S^2.

0, 1, 0, 1, 0, 1, 2, 1, 4, 1, 6, 2, 8, 7, 10, 16, 12, 29, 18, 46, 36, 67, 74, 93, 140, 136, 242, 224, 388, 401, 592, 727, 900, 1278, 1422, 2147, 2364, 3467, 4060, 5491, 7004, 8736, 11890, 14191, 19724, 23589, 32128, 39744, 51964, 66991, 84406, 111930, 138588

Offset

0,7

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).

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1)

Formula

G.f.: -((x (1 + x)^2 (1 - x + x^2 - x^3 + x^4)^2)/((-1 + x + x^6) (1 + x + x^6))).

a(n) = a(n-2) + 2*a(n-7) + a(n-12) for n >= 13.

Mathematica

z = 60; s = x + x^4; p = 1 - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292403 *)
LinearRecurrence[{0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1}, {0, 1, 0, 1, 0, 1, 2, 1, 4, 1, 6, 2}, 60] (* Vincenzo Librandi, Oct 01 2017 *)

Prog

(Magma) I:=[0, 1, 0, 1, 0, 1, 2, 1, 4, 1, 6, 2]; [n le 12 select I[n] else Self(n-2)+2*Self(n-7)+Self(n-12): n in [1..60]]; // Vincenzo Librandi, Oct 01 2017

Crossrefs

Cf. A292402.

Sequence in context: A131755 A305812 A341857 * A271773 A339602 A277127

Adjacent sequences: A292400 A292401 A292402 * A292404 A292405 A292406

Keyword

nonn,easy

Author

Clark Kimberling, Sep 30 2017

Status

approved