A389373 Expansion of g.f.: x/((1-x^2)^3*(1-2*x)).

0, 1, 2, 7, 14, 34, 68, 146, 292, 599, 1198, 2417, 4834, 9696, 19392, 38820, 77640, 155325, 310650, 621355, 1242710, 2485486, 4970972, 9942022, 19884044, 39768179, 79536358, 159072821, 318145642, 636291404, 1272582808, 2545165752, 5090331504, 10180663161, 20361326322, 40722652815

Offset

0,3

Comments

Equals the powers of 2 convolved with the aerated triangular numbers, [1, 0, 3, 0, 6, 0, 10, ...].

a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3) - 3*a(n-4) + 6*a(n-5) + a(n-6) - 2*a(n-7), n >= 8.

Links

Table of n, a(n) for n=0..35.

Formula

a(n) = (2^(n+9) - 18*n^2 - 168*n - 406)/432, n odd.

a(n) = (2^(n+9) - 36*n^2 - 264*n - 512)/432, n even.

Mathematica

CoefficientList[Series[x/((1-x^2)^3 (1-2x)), {x, 0, 40}], x] (* Harvey P. Dale, Dec 27 2025 *)

Crossrefs

Cf. A000079, A000217, A038163, A052161.

Sequence in context: A286861 A290682 A000147 * A334069 A128902 A227213

Adjacent sequences: A389370 A389371 A389372 * A389374 A389375 A389376

Keyword

nonn

Author

Enrique Navarrete, Nov 02 2025

Extensions

Zero prepended in Data field by Harvey P. Dale, Dec 28 2025

Offset changed to match insertion of 0 by Sean A. Irvine, Mar 01 2026

Status

approved