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A350897
Number of partitions of n such that 5*(smallest part) = (number of parts).
3
0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 55, 65, 74, 87, 99, 115, 131, 151, 172, 199, 226, 260, 298, 343, 393, 454, 522, 603, 696, 804, 929, 1076, 1243, 1438, 1664, 1924, 2222, 2567, 2961, 3413, 3931, 4520, 5193, 5959, 6827, 7811, 8928, 10186, 11607, 13208, 15008, 17028, 19297
OFFSET
1,7
COMMENTS
In general, for m >= 1, if g.f.= Sum_{k>=1} x^(m*k^2)/Product_{j=1..m*k-1} (1-x^j), then a(n) ~ r^2 * (m*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((m*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*m*(m - (m-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^m. - Vaclav Kotesovec, Oct 14 2024
LINKS
FORMULA
G.f.: Sum_{k>=1} x^(5*k^2)/Product_{j=1..5*k-1} (1-x^j).
a(n) ~ c * exp(Pi*sqrt(r*n)) / n^(3/4), where r = 0.42067169741517... and c = 0.04778365700734... - Vaclav Kotesovec, Jan 26 2022
a(n) ~ r^2 * (5*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((5*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(5*Pi*(5 - 3*r^2)) * n^(3/4)), where r = 0.808730600479392... is the real root of the equation r^2 = 1 - r^5. - Vaclav Kotesovec, Oct 14 2024
MATHEMATICA
CoefficientList[Series[Sum[x^(5k^2)/Product[1-x^j, {j, 5k-1}], {k, 62}], {x, 0, 62}], x] (* Stefano Spezia, Jan 22 2022 *)
PROG
(PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, sqrtint(N\5), x^(5*k^2)/prod(j=1, 5*k-1, 1-x^j))))
CROSSREFS
Column 5 of A350889.
Cf. A168657.
Sequence in context: A026810 A001400 A372703 * A008773 A008772 A351003
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 21 2022
STATUS
approved