A292402 -id:A292402 - cp's OEIS frontend

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

Clark Kimberling, Sep 30 2017

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).

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)

Cf. A292402.

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

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 *)

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.

A292404 p-INVERT of (1,0,0,1,0,0,0,0,0,0,...), where p(S) = 1 - S^4.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 4, 1, 0, 6, 8, 1, 4, 28, 12, 2, 56, 66, 16, 71, 220, 120, 76, 496, 560, 218, 816, 1821, 1148, 1200, 4396, 4847, 2816, 8386, 15536, 11122, 14716, 39256, 42760, 33346, 82480, 135292, 109760, 161931, 353256, 385528, 369380, 794378, 1198288

Offset: 0

Clark Kimberling, Sep 30 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 1, 0, 0, 4, 0, 0, 6, 0, 0, 4, 0, 0, 1)

Cf. A292402.

z = 60; s = x + x^4; p = 1 - s^4;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292404 *)

G.f.: -((x^3 (1 + x)^4 (1 - x + x^2)^4)/((-1 + x + x^4) (1 + x + x^4) (1 + x^2 + 2 x^5 + x^8))).

a(n) = a(n-4) + 4*a(n-7) + 6*a(n-10) + 4*a(n-13) + a(n-16) for n >= 17.

cp's OEIS Frontend

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

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