A373219 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 5^j)).

1, 1, 1, 2, 2, 4, 5, 6, 8, 10, 14, 17, 21, 26, 32, 42, 50, 60, 74, 88, 109, 130, 154, 184, 218, 263, 309, 363, 428, 500, 593, 690, 802, 936, 1084, 1266, 1463, 1687, 1948, 2242, 2590, 2970, 3403, 3900, 4456, 5107, 5820, 6626, 7546, 8572, 9755, 11059, 12523, 14180, 16028, 18133, 20455, 23056, 25978

Offset

0,4

Links

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

Vaclav Kotesovec, Plot of a(n) / (exp(Pi*sqrt(5*n/3)/2) / n^(3/4 + log(2)/(4*log(5)))) for n = 1..40000

Formula

G.f.: Product_{k>=1} (1 + x^k)^A055457(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^5).

a(n) ~ 5^(1/4) * Pi^(log(2)/(2*log(5))) * exp(Pi*sqrt(5*n/3)/2) / (2^(2 + (5*log(2) + 2*log(Pi))/(4*log(5))) * 3^(1/4 + log(2)/(4*log(5))) * n^(3/4 + log(2)/(4*log(5)))). - Vaclav Kotesovec, Mar 07 2026

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 + x^k)^(IntegerExponent[k, 5] + 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, 5)+1)))

Crossrefs

Cf. A000041, A327726, A174065, A373220, A373221, A393565.

Cf. A000009, A055457.

Sequence in context: A340445 A302486 A266745 * A062436 A342514 A238571

Adjacent sequences: A373216 A373217 A373218 * A373220 A373221 A373222

Keyword

nonn

Author

Seiichi Manyama, May 28 2024

Status

approved