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A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.

%I A363067 #27 Jun 19 2025 02:31:09
%S A363067 1,0,0,0,0,1,1,2,3,5,7,10,13,18,23,31,39,51,64,81,102,128,159,198,245,
%T A363067 304,374,460,563,689,841,1023,1242,1505,1819,2195,2642,3173,3804,4551,
%U A363067 5435,6477,7707,9151,10850,12843,15175,17902,21089,24802,29132,34164,40012,46796,54663,63766
%N A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.
%H A363067 Vaclav Kotesovec, <a href="/A363067/b363067.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Seiichi Manyama)
%F A363067 G.f.: Sum_{k>=0} x^(5*k)/Product_{j=1..4*k} (1-x^j).
%F A363067 a(n) ~ Gamma(1/4) * Pi^(1/4) * exp(Pi*sqrt(2*n/3)) / (2^(49/8) * 3^(5/8) * n^(9/8)). - _Vaclav Kotesovec_, Jun 19 2025
%e A363067 a(8) = 3 counts these partitions:  431, 4211, 41111.
%t A363067 nmax = 60; CoefficientList[Series[Sum[x^(5*k)/Product[1 - x^j, {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* _Vaclav Kotesovec_, Jun 18 2025 *)
%t A363067 nmax = 60; p=1; s=1; Do[p=Expand[p*(1-x^(4*k))*(1-x^(4*k-1))*(1-x^(4*k-2))*(1-x^(4*k-3))]; p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]]; s+=x^(5*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 18 2025 *)
%o A363067 (PARI) a(n) = sum(k=0, n\5, #partitions(n-5*k, 4*k));
%Y A363067 Cf. A002865, A238479, A363066, A363068.
%Y A363067 Cf. A237826, A363046.
%Y A363067 Cf. A035296.
%K A363067 nonn
%O A363067 0,8
%A A363067 _Seiichi Manyama_, May 16 2023