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A348163
Number of partitions of n such that 4*(greatest part) = (number of parts).
3
0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 4, 5, 6, 8, 9, 12, 14, 16, 18, 22, 25, 30, 35, 42, 49, 60, 68, 81, 93, 109, 127, 149, 171, 200, 231, 269, 309, 359, 410, 474, 544, 625, 715, 824, 939, 1080, 1232, 1411, 1607, 1839, 2090, 2385, 2708, 3081, 3493, 3972, 4493
OFFSET
1,14
COMMENTS
Also, the number of partitions of n such that (greatest part) = 4*(number of parts).
LINKS
FORMULA
G.f.: Sum_{k>=1} x^(5*k-1) * Product_{j=1..k-1} (1-x^(4*k+j-1))/(1-x^j).
a(n) ~ Pi^4 * exp(Pi*sqrt(2*n/3)) / (2*3^(3/2)*n^3). - Vaclav Kotesovec, Oct 17 2024
EXAMPLE
a(16) = 3 counts these partitions:
[3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[2, 2, 2, 2, 2, 2, 2, 2].
MATHEMATICA
nmax = 100; Rest[CoefficientList[Series[Sum[x^(5*k-1) * Product[(1 - x^(4*k+j-1)) / (1 - x^j), {j, 1, k-1}], {k, 1, nmax/5 + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 15 2024 *)
nmax = 100; p = x^3; s = x^3; Do[p = Normal[Series[p*x^5*(1 - x^(5*k - 1))*(1 - x^(5*k))*(1 - x^(5*k + 1))*(1 - x^(5*k + 2))*(1 - x^(5*k + 3))/((1 - x^(4*k + 3))*(1 - x^(4*k + 2))*(1 - x^(4*k + 1))*(1 - x^(4*k))*(1 - x^k)), {x, 0, nmax}]]; s += p; , {k, 1, nmax/5 + 1}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Oct 16 2024 *)
PROG
(PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(5*k-1)*prod(j=1, k-1, (1-x^(4*k+j-1))/(1-x^j)))))
CROSSREFS
Column 4 of A350879.
Sequence in context: A319611 A337102 A239495 * A298210 A045772 A294977
KEYWORD
nonn,changed
AUTHOR
Seiichi Manyama, Jan 25 2022
STATUS
approved