A292480 p-INVERT of the odd positive integers, where p(S) = 1 - S^2.

0, 1, 6, 20, 56, 160, 480, 1456, 4384, 13136, 39360, 118064, 354272, 1062928, 3188736, 9565936, 28697632, 86093264, 258280512, 774841520, 2324523104, 6973567888, 20920705152, 62762119792, 188286360736, 564859074896, 1694577214656, 5083731648560

Offset

0,3

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).

In the following guide to p-INVERT sequences using s = (1,3,5,7,9,...) = A005408, in some cases t(1,3,5,7,9,...) is a shifted (or differently indexed) version of the cited sequence:

p(S) *********** t(1,3,5,7,9,...)

1 - S A003946

1 - S^2 A292480

1 - S^3 (not yet in OEIS)

(1 - S)^2 (not yet in OEIS)

(1 - S)^3 (not yet in OEIS)

1 - S - S^2 A289786

1 + S - S^2 A289484

1 - S - 2 S^2 A289785

1 - S - 3 S^2 A289786

1 - S - 4 S^2 A289787

1 - S - 5 S^2 A289788

1 - S - 6 S^2 A289789

1 - S - 7 S^2 A289790

1 + S - 2 S^2 A289791

1 - S + S^2 - S^3 A289792

1 + S - 3 S^2 A289793

1 - S - S^2 - S^3 A289794

Links

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

Index entries for linear recurrences with constant coefficients, signature (4,-5,6)

Formula

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

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

Example

s = (1,3,5,7,9,...), S(x) = x + 3 x^2 + 5 x^3 + 7 x^4 + ...,
p(S(x)) = 1 - ( x + 3 x^2 + 5 x^3 + 7 x^4 + ...)^2,
1/p(S(x)) = 1 + x^2 + 6 x^3 + 20 x^4 + 56 x^5 + ...
T(x) = (-1 + 1/p(S(x)))/x = x + 6 x^2 + 20 x^3 + 56 x^4 + ...
t(s) = (0,1,2,20,56,...).

Mathematica

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292480 *)
Join[{0}, LinearRecurrence[{4, -5, 6}, {1, 6, 20}, 30]] (* Vincenzo Librandi, Oct 03 2017 *)

Prog

(Magma) I:=[0, 1, 6, 20]; [n le 4 select I[n] else 4*Self(n-1)- 5*Self(n-2)+6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 03 2017

Crossrefs

Cf. A005408, A292479.

Sequence in context: A014480 A048778 A048611 * A200528 A127982 A109164

Adjacent sequences: A292477 A292478 A292479 * A292481 A292482 A292483

Keyword

nonn,easy

Author

Clark Kimberling, Oct 02 2017

Status

approved