A350892 Number of partitions of n such that 3*(smallest part) = (number of parts).

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 12, 15, 18, 22, 27, 33, 40, 48, 58, 69, 82, 98, 115, 135, 158, 184, 214, 248, 286, 330, 379, 435, 497, 569, 648, 739, 840, 955, 1082, 1228, 1388, 1572, 1775, 2005, 2259, 2549, 2867, 3228, 3626, 4076, 4571, 5131, 5745, 6438, 7199, 8053, 8992, 10045, 11199

Offset

1,5

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..20000

Formula

G.f.: Sum_{k>=1} x^(3*k^2)/Product_{j=1..3*k-1} (1-x^j).

a(n) ~ c * exp(2*sqrt((5*log(A075778)^2 + 2*polylog(2, 1 - A075778))*n)) / n^(3/4), where c = 0.0582980106266835787... - Vaclav Kotesovec, Jan 24 2022

Mathematica

CoefficientList[Series[Sum[x^(3k^2)/Product[1-x^j, {j, 3k-1}], {k, 64}], {x, 0, 64}], x] (* Stefano Spezia, Jan 22 2022 *)
Table[Count[IntegerPartitions[n], _?(3#[[-1]]==Length[#]&)], {n, 70}] (* Harvey P. Dale, Jul 13 2023 *)

Prog

(PARI) my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, sqrtint(N\3), x^(3*k^2)/prod(j=1, 3*k-1, 1-x^j))))

Crossrefs

Column 3 of A350889.

Cf. A168657, A237756, A350894.

Sequence in context: A210717 A171962 A238208 * A029028 A240572 A029072

Adjacent sequences: A350889 A350890 A350891 * A350893 A350894 A350895

Keyword

nonn

Author

Seiichi Manyama, Jan 21 2022

Status

approved