A292479 p-INVERT of the positive squares, where p(S) = 1 - S^2.

0, 1, 8, 35, 120, 392, 1336, 4725, 16792, 59191, 207536, 727440, 2553264, 8968569, 31502248, 110627195, 388451624, 1364010648, 4789766120, 16819647565, 59063332152, 207403715119, 728306773600, 2557481457440, 8980717116000, 31536219644721, 110740934436168

Offset: 0

Author

Clark Kimberling, Oct 02 2017

Keywords

• nonn
• easy

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 = (1,4,9,16,...) = A000290, in some cases t(1,4,9,16,...) is a shifted (or differently indexed) version of the cited sequence:

** p(S) ********** t(1, 4, 9, 16,...)

1 - S A033453

1 - S^2 A292479

1 - S^3 (not yet in OEIS)

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

1 - S - S^2 A289779

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

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

1 + S - 3 S^2 (not yet in OEIS)

Examples

			s = (1,4,9,16,25,...), S(x) = x + 4 x^2 + 9 x^3 + 16 x^4 + ...,
p(S(x)) = 1 - (x + 4 x^2 + 9 x^3 + 16 x^4 + ...)^2,
1/p(S(x)) = 1 + x^2 + 8*x^3 + 35*x^4 + 120*x^5 + ...
T(x) = (-1 + 1/p(S(x)))/x = x + 8 x^2 + 35 x^3 + 120 x^4 + ...
t(s) = (0, 1, 8, 35, 120, ...).
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-14,22,-14,6,-1)

Crossrefs

Cf. A000290, A292480.

Programs

Magma

I:=[0,1,8,35,120,392]; [n le 6 select I[n] else 6*Self(n-1)-14*Self(n-2)+22*Self(n-3)-14*Self(n-4)+6*Self(n-5)- Self(n-6): n in [1..30]]; // Vincenzo Librandi, Oct 03 2017

Mathematica

z = 60; s = x (x + 1)/(1 - x)^3; p = 1 - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000290 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292479 *)
LinearRecurrence[{6, -14, 22, -14, 6, -1}, {0, 1, 8, 35, 120, 392}, 30] (* Vincenzo Librandi, Oct 03 2017 *)

Formula

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

a(n) = 6*a(n-1) - 14*a(n-2) + 22*a(n-3) - 14*a(n-4) + 6*a(n-5) - a(n-6) for n >= 7.