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A361459 Number of partitions p of n such that 5*min(p) is a part of p.

%I A361459 #27 Jun 19 2025 09:29:04
%S A361459 0,0,0,0,0,1,1,2,3,5,7,12,15,23,31,44,58,82,105,142,185,244,312,409,
%T A361459 516,664,837,1063,1328,1674,2074,2588,3194,3952,4847,5964,7270,8884,
%U A361459 10786,13104,15832,19147,23027,27709,33203,39776,47476,56661,67382,80108,94960,112494,132919,156965
%N A361459 Number of partitions p of n such that 5*min(p) is a part of p.
%C A361459 From _Vaclav Kotesovec_, Jun 19 2025: (Start)
%C A361459 In general, for m>1, if g.f. = Sum_{k>=0} x^(m*k) / Product_{j>=k} (1 - x^j), then A000041(n) - a(n) ~ Pi * (m-1) * exp(Pi*sqrt(2*n/3)) / (3*2^(5/2)*n^(3/2)).
%C A361459 a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 - (sqrt(3/2)/Pi + (m - 23/24)*Pi / sqrt(6)) / sqrt(n)). (End)
%H A361459 Vaclav Kotesovec, <a href="/A361459/b361459.txt">Table of n, a(n) for n = 1..10000</a>
%F A361459 G.f.: Sum_{k>=1} x^(6*k)/Product_{j>=k} (1-x^j).
%F A361459 From _Vaclav Kotesovec_, Jun 19 2025: (Start)
%F A361459 a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 121*Pi/(24*sqrt(6))) / sqrt(n)).
%F A361459 A000041(n) - a(n) ~ 5 * Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). (End)
%p A361459 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A361459       `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
%p A361459     end:
%p A361459 a:= n-> add(b(n-6*i, i), i=1..n/6):
%p A361459 seq(a(n), n=1..60);  # _Alois P. Heinz_, May 17 2023
%t A361459 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i+1] + b[n-i, i]]];
%t A361459 a[n_] := Sum[b[n - 6 i, i], {i, 1, n/6}];
%t A361459 Array[a, 60] (* _Jean-François Alcover_, May 30 2024, after _Alois P. Heinz_ *)
%o A361459 (PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0, 0], Vec(sum(k=1, N, x^(6*k)/prod(j=k, N, 1-x^j))))
%Y A361459 Cf. A117989, A238589, A238590, A238591.
%Y A361459 Cf. A237827, A363068.
%K A361459 nonn
%O A361459 1,8
%A A361459 _Seiichi Manyama_, May 17 2023