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A363068 Number of partitions p of n such that (1/5)*max(p) is a part of p.

%I A363068 #28 Jun 19 2025 03:13:56
%S A363068 1,0,0,0,0,0,1,1,2,3,5,7,11,14,20,26,35,44,59,73,94,117,148,181,228,
%T A363068 277,344,418,514,621,762,917,1116,1342,1624,1945,2348,2803,3366,4012,
%U A363068 4798,5700,6798,8052,9565,11305,13383,15771,18618,21880,25745,30187,35414,41414,48461,56531,65967
%N A363068 Number of partitions p of n such that (1/5)*max(p) is a part of p.
%C A363068 In general, for m>=1, if g.f. = Sum_{k>=0} x^((m+1)*k) / Product_{j=1..m*k} (1 - x^j), then a(n) ~ Gamma(1/m) * Pi^(1/m) * exp(Pi*sqrt(2*n/3)) / (m^2 * 2^((4*m+1)/(2*m)) * 3^((m+1)/(2*m)) * n^(1 + 1/(2*m))). - _Vaclav Kotesovec_, Jun 19 2025
%H A363068 Vaclav Kotesovec, <a href="/A363068/b363068.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..1000 from Seiichi Manyama)
%F A363068 G.f.: Sum_{k>=0} x^(6*k)/Product_{j=1..5*k} (1-x^j).
%F A363068 a(n) ~ Gamma(1/5) * Pi^(1/5) * exp(Pi*sqrt(2*n/3)) / (25 * 2^(21/10) * 3^(3/5) * n^(11/10)). - _Vaclav Kotesovec_, Jun 19 2025
%e A363068 a(8) = 2 counts these partitions:  521, 5111.
%t A363068 nmax = 60; CoefficientList[Series[Sum[x^(6*k)/Product[1 - x^j, {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* _Vaclav Kotesovec_, Jun 18 2025 *)
%t A363068 nmax = 60; p=1; s=1; Do[p=Expand[p*(1-x^(5*k))*(1-x^(5*k-1))*(1-x^(5*k-2))*(1-x^(5*k-3))*(1-x^(5*k-4))]; p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]]; s+=x^(6*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 18 2025 *)
%o A363068 (PARI) a(n) = sum(k=0, n\6, #partitions(n-6*k, 5*k));
%Y A363068 Cf. A002865, A238479, A363066, A363067.
%Y A363068 Cf. A237827, A363047.
%Y A363068 Cf. A035297, A035298.
%K A363068 nonn
%O A363068 0,9
%A A363068 _Seiichi Manyama_, May 16 2023