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A350892
Number of partitions of n such that 3*(smallest part) = (number of parts).
3
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
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 = (3*log(A075778)^2 + polylog(2, A075778^2))^(1/4) / (2*sqrt(3*Pi*(1 + A075778)*(2 + 3*A075778))) = 0.0582980106266835787... - Vaclav Kotesovec, Jan 24 2022, updated Oct 14 2024
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.
Sequence in context: A210717 A171962 A238208 * A029028 A240572 A029072
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 21 2022
STATUS
approved