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A348163 Number of partitions of n such that 4*(greatest part) = (number of parts).

%I A348163 #26 Oct 18 2024 03:54:39
%S A348163 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,
%T A348163 35,42,49,60,68,81,93,109,127,149,171,200,231,269,309,359,410,474,544,
%U A348163 625,715,824,939,1080,1232,1411,1607,1839,2090,2385,2708,3081,3493,3972,4493
%N A348163 Number of partitions of n such that 4*(greatest part) = (number of parts).
%C A348163 Also, the number of partitions of n such that (greatest part) = 4*(number of parts).
%H A348163 Vaclav Kotesovec, <a href="/A348163/b348163.txt">Table of n, a(n) for n = 1..10000</a>
%F A348163 G.f.: Sum_{k>=1} x^(5*k-1) * Product_{j=1..k-1} (1-x^(4*k+j-1))/(1-x^j).
%F A348163 a(n) ~ Pi^4 * exp(Pi*sqrt(2*n/3)) / (2*3^(3/2)*n^3). - _Vaclav Kotesovec_, Oct 17 2024
%e A348163 a(16) = 3 counts these partitions:
%e A348163 [3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
%e A348163 [3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1],
%e A348163 [2, 2, 2, 2, 2, 2, 2, 2].
%t A348163 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 *)
%t A348163 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 *)
%o A348163 (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)))))
%Y A348163 Column 4 of A350879.
%Y A348163 Cf. A350896, A350898.
%K A348163 nonn
%O A348163 1,14
%A A348163 _Seiichi Manyama_, Jan 25 2022