A373221 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 7^j)).

1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 14, 18, 22, 28, 34, 41, 50, 60, 72, 86, 105, 124, 146, 174, 204, 240, 282, 332, 386, 450, 524, 606, 703, 812, 940, 1082, 1243, 1428, 1636, 1873, 2140, 2448, 2788, 3172, 3610, 4096, 4646, 5264, 5963, 6737, 7607, 8584, 9668, 10887, 12244, 13765, 15451, 17328

Offset

0,4

Comments

This sequence is different from A093950. The first difference occurs at a(50) = 6737, whereas A093950(50) = 6736.

From Vaclav Kotesovec, Feb 21 2026: (Start)

In general, for m > 1, if g.f. = Product_{i>=1, j>=0} (1 + x^(i * m^j)), then log(a(n)) ~ Pi*sqrt(m*n/(3*(m-1))).

More precisely, a(n) ~ c(m) * exp(Pi*sqrt(m*n/(3*(m-1)))) / n^(3/4 + log(2)/(4*log(m))), where c(m) are constants.

c(2) = 1/(4*sqrt(3)), c(4) = 1 / (2^(11/8) * 3^(3/4)), c(8) = 1 / (2 * 21^(1/3)). (End)

c(m) = m^(1/4) * Pi^(log(2)/(2*log(m))) / (2^(3/2 + (3*log(2) + 2*log(Pi))/(4*log(m))) * (3*(m-1))^(1/4 + log(2)/(4*log(m)))). - Vaclav Kotesovec, Mar 07 2026

a(n) is the number of partitions of n into distinct parts of the form i * 7^j (for i>=1, j>=0). - Joerg Arndt, Mar 07 2026

Links

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

Vaclav Kotesovec, Plot of a(n) / (exp(Pi*sqrt(7*n/2)/3) / n^(3/4 + log(2)/(4*log(7)))) for n = 1..80000

Formula

G.f.: Product_{k>=1} (1 + x^k)^A373217(k).

Let A(x) be the g.f. of this sequence, and B(x) be the g.f. of A000009, then B(x) = A(x)/A(x^7).

a(n) ~ 7^(1/4) * Pi^(log(2)/(2*log(7))) * exp(Pi*sqrt(7*n/2)/3) / (2^(7/4 + (2*log(2) + log(Pi))/(2*log(7))) * 3^(1/2 + log(2)/(2*log(7))) * n^(3/4 + log(2)/(4*log(7)))). - Vaclav Kotesovec, Mar 07 2026

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 + x^k)^(IntegerExponent[k, 7] + 1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 20 2026 *)

Prog

(PARI) my(N=60, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(valuation(k, 7)+1)))

Crossrefs

Cf. A000041 (m=2), A327726 (m=3), A174065 (m=4), A373219 (m=5), A373220 (m=6), A393565 (m=8).

Cf. A000009, A093950, A373217.

Sequence in context: A266747 A246762 A093950 * A280715 A023894 A285799

Adjacent sequences: A373218 A373219 A373220 * A373222 A373223 A373224

Keyword

nonn

Author

Seiichi Manyama, May 28 2024

Status

approved