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A348164 Number of partitions of n such that 5*(greatest part) = (number of parts).

%I A348164 #24 Oct 18 2024 03:54:23
%S A348164 0,0,0,0,1,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,5,5,7,8,10,11,15,16,20,
%T A348164 22,26,28,35,38,46,52,62,70,85,95,112,127,148,166,195,219,254,288,332,
%U A348164 375,435,489,562,635,726,817,936,1051,1198,1348,1531,1721,1957,2196,2489
%N A348164 Number of partitions of n such that 5*(greatest part) = (number of parts).
%C A348164 Also, the number of partitions of n such that (greatest part) = 5*(number of parts).
%H A348164 Vaclav Kotesovec, <a href="/A348164/b348164.txt">Table of n, a(n) for n = 1..10000</a>
%F A348164 G.f.: Sum_{k>=1} x^(6*k-1) * Product_{j=1..k-1} (1-x^(5*k+j-1))/(1-x^j).
%F A348164 a(n) ~ 5 * Pi^5 * exp(Pi*sqrt(2*n/3)) / (9 * 2^(3/2) * n^(7/2)). - _Vaclav Kotesovec_, Oct 17 2024
%e A348164 a(19) = 3 counts these partitions:
%e A348164 [3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
%e A348164 [3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
%e A348164 [2, 2, 2, 2, 2, 2, 2, 2, 2, 1].
%t A348164 nmax = 100; Rest[CoefficientList[Series[Sum[x^(6*k-1) * Product[(1 - x^(5*k+j-1)) / (1 - x^j), {j, 1, k-1}], {k, 1, nmax/6 + 1}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Oct 15 2024 *)
%t A348164 nmax = 100; p = x^4; s = x^4; Do[p = Normal[Series[p*x^6*(1 - x^(6*k - 1))*(1 - x^(6*k))*(1 - x^(6*k + 1))*(1 - x^(6*k + 2))*(1 - x^(6*k + 3))*(1 - x^(6*k + 4))/((1 - x^(5*k + 4))*(1 - x^(5*k + 3))*(1 - x^(5*k + 2))*(1 - x^(5*k + 1))*(1 - x^(5*k))*(1 - x^k)), {x, 0, nmax}]]; s += p;, {k, 1, nmax/6 + 1}]; Take[CoefficientList[s, x], nmax] (* _Vaclav Kotesovec_, Oct 16 2024 *)
%o A348164 (PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(6*k-1)*prod(j=1, k-1, (1-x^(5*k+j-1))/(1-x^j)))))
%Y A348164 Column 5 of A350879.
%Y A348164 Cf. A350897, A350899.
%K A348164 nonn
%O A348164 1,17
%A A348164 _Seiichi Manyama_, Jan 25 2022