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A290917 p-INVERT of the positive integers, where p(S) = (1 - S)^2. 2
2, 7, 22, 67, 200, 588, 1708, 4913, 14018, 39725, 111922, 313752, 875702, 2434747, 6746350, 18636343, 51340988, 141089508, 386857888, 1058572325, 2891193242, 7882921697, 21458980582, 58330331952, 158339542250, 429274563823, 1162435429318, 3144299295403 (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 A290890 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 (6, -11, 6, -1)

FORMULA

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

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).

(a(n)) is the p-INVERT of (1,1,1,1,1...) using p(S) = (1 - S - S^2)^2.

a(n) = (((3-sqrt(5))/2)^n * (-3+sqrt(5)) * (-5+7*sqrt(5)-5*n) + 2^(-n) * (3+sqrt(5))^(n+1) * (5+7*sqrt(5)+5*n)) / 50. - Colin Barker, Aug 24 2017

MATHEMATICA

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

Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)

Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290917 *)

PROG

(PARI) Vec((2 - x)*(1 - 2*x) / (1 - 3*x + x^2)^2 + O(x^30)) \\ Colin Barker, Aug 24 2017

CROSSREFS

Cf. A000027, A290890.

Sequence in context: A084264 A088211 A071684 * A060816 A171847 A037552

Adjacent sequences:  A290914 A290915 A290916 * A290918 A290919 A290920

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 18 2017

STATUS

approved

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Last modified October 17 12:13 EDT 2019. Contains 328111 sequences. (Running on oeis4.)