login
A350896
Number of partitions of n such that 4*(smallest part) = (number of parts).
3
0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 20, 22, 26, 30, 35, 40, 48, 55, 65, 76, 90, 105, 126, 147, 175, 206, 244, 286, 339, 396, 467, 545, 638, 741, 865, 1000, 1160, 1337, 1543, 1770, 2035, 2325, 2660, 3029, 3451, 3916, 4447, 5029, 5691, 6419, 7242, 8146, 9167, 10286, 11546, 12930, 14481, 16185
OFFSET
1,6
LINKS
FORMULA
G.f.: Sum_{k>=1} x^(4*k^2)/Product_{j=1..4*k-1} (1-x^j).
a(n) ~ c * exp(Pi*sqrt(2*n/5)) / n^(3/4), where c = (3 - sqrt(5))^(1/4) / (8*sqrt(5)) = 0.05226232058... - Vaclav Kotesovec, Jan 25 2022, updated Oct 13 2024
EXAMPLE
For n=7 there are a(7)=3 such partitions: [1,2,2,2], [1,1,2,3] and [1,1,1,4]. - R. J. Mathar, Jun 20 2022
MATHEMATICA
CoefficientList[Series[Sum[x^(4k^2)/Product[1-x^j, {j, 4k-1}], {k, 63}], {x, 0, 63}], x] (* Stefano Spezia, Jan 22 2022 *)
PROG
(PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, sqrtint(N\4), x^(4*k^2)/prod(j=1, 4*k-1, 1-x^j))))
CROSSREFS
Column 4 of A350889.
Cf. A168657.
Sequence in context: A008755 A029006 A085756 * A008754 A029005 A132154
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 21 2022
STATUS
approved