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A238590 Number of partitions p of n such that 3*min(p) is a part of p.

%I A238590 #24 Jun 19 2025 03:53:26
%S A238590 0,0,0,1,1,2,3,6,7,12,16,25,32,46,61,86,110,149,192,257,326,425,538,
%T A238590 694,871,1107,1381,1740,2154,2689,3313,4103,5024,6176,7529,9201,11157,
%U A238590 13554,16365,19784,23782,28610,34260,41039,48958,58405,69431,82525,97775
%N A238590 Number of partitions p of n such that 3*min(p) is a part of p.
%H A238590 Vaclav Kotesovec, <a href="/A238590/b238590.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F A238590 G.f.: Sum_{k>=1} x^(4*k)/Product_{j>=k} (1-x^j). - _Seiichi Manyama_, May 17 2023
%F A238590 From _Vaclav Kotesovec_, Jun 19 2025: (Start)
%F A238590 a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 73*Pi/(24*sqrt(6))) / sqrt(n)).
%F A238590 A000041(n) - a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (2^(5/2) * n^(3/2)). (End)
%e A238590 a(7) = 3 counts these partitions:  331, 3211, 31111.
%p A238590 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A238590       `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
%p A238590     end:
%p A238590 a:= n-> add(b(n-4*i, i), i=1..n/4):
%p A238590 seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 03 2014
%t A238590 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 3*Min[p]]], {n, 50}]
%t A238590 (* Second program: *)
%t A238590 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
%t A238590 a[n_] := Sum[b[n-4i, i], {i, 1, n/4}];
%t A238590 Array[a, 60] (* _Jean-François Alcover_, Jun 04 2021, after _Alois P. Heinz_ *)
%o A238590 (PARI) my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=k, N, 1-x^j)))) \\ _Seiichi Manyama_, May 17 2023
%Y A238590 Cf. A117989, A238589, A238591.
%Y A238590 Cf. A237825, A363066.
%K A238590 nonn,easy
%O A238590 1,6
%A A238590 _Clark Kimberling_, Mar 01 2014