A291024 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - 2 S^2)^2.

0, 4, 8, 24, 64, 172, 456, 1200, 3136, 8148, 21064, 54216, 139008, 355196, 904840, 2298720, 5825408, 14729636, 37168008, 93612408, 235369664, 590852172, 1481051720, 3707411472, 9268764096, 23145174388, 57732471752, 143857070376, 358113876352, 890666303260

Offset: 0

Author

Clark Kimberling, Aug 24 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).

See A291000 for a guide to related sequences.

Links

• Clark Kimberling, Table of n, a(n) for n = 0..999
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Crossrefs

Cf. A000012, A033453, A289780, A291000, A291142.

Programs

Mathematica

z = 60; s = x/(1 - x); p = 1 - 3 s^2 + 2 s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291024 *)
u/4 (* A291142 *)

PARI

concat(0, Vec(4*x*(1 - 2*x) / (1 - 2*x - x^2)^2 + O(x^30))) \\ Colin Barker, Aug 24 2017

Formula

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

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

a(n) = 4*A291142(n) for n >= 0.

a(n) = ((1+sqrt(2))^n*(3*sqrt(2) + 2*(-1+sqrt(2))*n) - (1-sqrt(2))^n*(3*sqrt(2) + 2*(1+sqrt(2))*n)) / 4. - Colin Barker, Aug 24 2017

E.g.f.: exp(x)*x*cosh(sqrt(2)*x) + 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, Jun 07 2025