A308986 Expansion of Product_{k>=0} 1/(1 + 2*x^(2^k)).

1, -2, 2, -4, 10, -20, 36, -72, 154, -308, 596, -1192, 2420, -4840, 9608, -19216, 38586, -77172, 154036, -308072, 616740, -1233480, 2465768, -4931536, 9865492, -19730984, 39457128, -78914256, 157838120, -315676240, 631333264, -1262666528, 2525371642, -5050743284

Offset: 0

Author

Ilya Gutkovskiy, Jul 04 2019

Keywords

• sign

Comments

Convolution inverse of A001316.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..3322
• Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, Picturesque convolution-like recurrences and partial sums' generation, arXiv:2507.23619 [math.NT], 2025. See p. 28.

Crossrefs

Cf. A000120, A001316, A006046, A308985.

Programs

Mathematica

nmax = 33; CoefficientList[Series[Product[1/(1 + 2 x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[2^DigitCount[k, 2, 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} 2^A000120(k) * a(n-k).