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A363066 Number of partitions p of n such that (1/3)*max(p) is a part of p.

%I A363066 #33 Jun 29 2025 18:23:08
%S A363066 1,0,0,0,1,1,2,3,5,6,9,11,16,20,27,33,45,55,72,89,116,142,181,222,281,
%T A363066 343,429,522,649,786,967,1168,1429,1719,2088,2504,3026,3615,4345,5174,
%U A363066 6192,7349,8755,10360,12297,14507,17154,20182,23788,27910,32790,38374,44955,52480,61307,71402
%N A363066 Number of partitions p of n such that (1/3)*max(p) is a part of p.
%H A363066 Vaclav Kotesovec, <a href="/A363066/b363066.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Seiichi Manyama)
%F A363066 G.f.: Sum_{k>=0} x^(4*k)/Product_{j=1..3*k} (1-x^j).
%F A363066 a(n) ~ Gamma(1/3) * Pi^(1/3) * exp(Pi*sqrt(2*n/3)) / (2^(13/6) * 3^(8/3) * n^(7/6)). - _Vaclav Kotesovec_, Jun 19 2025
%e A363066 a(7) = 3 counts these partitions:  331, 3211, 31111.
%t A363066 nmax = 60; CoefficientList[Series[Sum[x^(4*k)/Product[1 - x^j, {j, 1, 3*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* _Vaclav Kotesovec_, Jun 18 2025 *)
%t A363066 nmax = 60; p=1; s=1; Do[p=Expand[p*(1-x^(3*k))*(1-x^(3*k-1))*(1-x^(3*k-2))]; p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]]; s+=x^(4*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 18 2025 *)
%t A363066 Join[{1},Table[Count[IntegerPartitions[n],_?(MemberQ[#,#[[1]]/3]&)],{n,60}]] (* _Harvey P. Dale_, Jun 29 2025 *)
%o A363066 (PARI) a(n) = sum(k=0, n\4, #partitions(n-4*k, 3*k));
%Y A363066 Cf. A002865, A238479, A363067, A363068.
%Y A363066 Cf. A008484, A237825, A363045.
%Y A363066 Cf. A035295.
%K A363066 nonn
%O A363066 0,7
%A A363066 _Seiichi Manyama_, May 16 2023