A292489 - OEIS

A292489

p-INVERT of the odd positive integers, where p(S) = 1 - S - 6 S^2.

1, 10, 60, 312, 1656, 8928, 48024, 257904, 1385352, 7442784, 39985272, 214811280, 1154025000, 6199749504, 33306803352, 178933509936, 961281138888, 5164272731808, 27743925989304, 149048175357648, 800728728609384, 4301739993919680, 23110157427289560

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OFFSET

0,2

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 A292480 for a guide to related sequences.

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (5, -1, 15, 6)

FORMULA

G.f.: -(((1 + x) (1 + 4 x + 7 x^2))/((-1 + 5 x + 2 x^2) (1 + 3 x^2))).

a(n) = 5*a(n-1) - a(n-2) + 16*a(n-3) + 6*a(n-4) for n >= 5.

MATHEMATICA

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s - 6 s^2;

Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)

Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292489 *)

PROG