A393565 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 8^j)).

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 14, 17, 21, 26, 32, 39, 47, 57, 68, 81, 97, 115, 136, 162, 190, 223, 262, 305, 356, 415, 481, 558, 646, 745, 859, 988, 1134, 1301, 1490, 1704, 1945, 2218, 2525, 2870, 3261, 3698, 4189, 4743, 5360, 6052, 6828, 7691, 8658, 9738, 10939

Offset

0,4

Comments

It differs from A280663 starting at a(27).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Vaclav Kotesovec, Plot of a(n) / (exp(Pi*sqrt(8*n/21)) / n^(5/6)) for n = 1..50000

Formula

a(n) ~ exp(Pi*sqrt(8*n/21)) / (2 * 21^(1/3) * n^(5/6)). - Vaclav Kotesovec, Mar 07 2026

Mathematica

nmax = 60; CoefficientList[Series[Product[(1+x^k)^(IntegerExponent[k, 8]+1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[Product[1 + x^(i*8^j), {i, 1, nmax}, {j, 0, Log[nmax]/Log[8] + 1}], {x, 0, nmax}], x]

Crossrefs

Cf. A000041, A327726, A174065, A373219, A373220, A373221.

Cf. A280663.

Sequence in context: A326491 A266748 A304883 * A280663 A052816 A122130

Adjacent sequences: A393562 A393563 A393564 * A393566 A393567 A393568

Keyword

nonn

Author

Vaclav Kotesovec, Feb 21 2026

Status

approved