A118442 Column 0 of triangle A118441, which is the matrix log of triangle A118435.

0, 1, -4, -12, 32, 80, -192, -448, 1024, 2304, -5120, -11264, 24576, 53248, -114688, -245760, 524288, 1114112, -2359296, -4980736, 10485760, 22020096, -46137344, -96468992, 201326592, 419430400, -872415232, -1811939328, 3758096384, 7784628224, -16106127360, -33285996544

Offset

0,3

Comments

Let M(n) be the nXn Hankel matrix with i-th row and j-th column entry abs(i + j - n - 1). The determinant of M(n) is a(n-1). - Michael Somos, Jan 25 2026

Links

Stefano Spezia, Table of n, a(n) for n = 0..3000

Index entries for linear recurrences with constant coefficients, signature (0,-8,0,-16).

Formula

a(n) = (-1)^floor(n/2)*A001787(n).

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

E.g.f.: x*(cos(2*x) - sin(2*x)). - Stefano Spezia, Jul 01 2023

Sum_{n>=1} 1/a(n) = 2*arctan(1/2) - log(5/4). - Amiram Eldar, Dec 03 2025

Mathematica

LinearRecurrence[{0, -8, 0, -16}, {0, 1, -4, -12}, 40] (* Harvey P. Dale, Dec 29 2014 *)
a[ n_] := If[n < 0, 0, (-1)^Floor[n/2] * n * 2^(n-1)]; (* Michael Somos, Jan 25 2026 *)

Prog

(PARI) {a(n)=polcoeff(x*(1 - 4*x - 4*x^2)/(1 + 4*x^2 +x*O(x^n))^2, n)}
(PARI) a(n)= n<<(n-1)*(-1)^(n\2); \\ Ruud H.G. van Tol, Nov 30 2025

Crossrefs

Cf. A118441 (triangle), A118443 (row sums), A118435, A001787, A073000.

Sequence in context: A260186 A085750 A001787 * A139756 A097067 A038592

Adjacent sequences: A118439 A118440 A118441 * A118443 A118444 A118445

Keyword

sign,easy

Author

Paul D. Hanna, Apr 28 2006

Status

approved