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A238591 Number of partitions p of n such that 4*min(p) is a part of p.

%I A238591 #23 Jun 19 2025 08:54:56
%S A238591 0,0,0,0,1,1,2,3,5,8,11,16,23,32,45,60,81,109,144,190,247,320,412,529,
%T A238591 675,854,1078,1355,1695,2117,2626,3251,4010,4932,6047,7394,9012,10959,
%U A238591 13290,16083,19407,23379,28090,33689,40317,48158,57406,68324,81155,96248
%N A238591 Number of partitions p of n such that 4*min(p) is a part of p.
%H A238591 Vaclav Kotesovec, <a href="/A238591/b238591.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F A238591 G.f.: Sum_{k>=1} x^(5*k)/Product_{j>=k} (1-x^j). - _Seiichi Manyama_, May 17 2023
%F A238591 From _Vaclav Kotesovec_, Jun 19 2025: (Start)
%F A238591 a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 97*Pi/(24*sqrt(6))) / sqrt(n)).
%F A238591 A000041(n) - a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3*sqrt(2)*n^(3/2)). (End)
%e A238591 a(9) = 5 counts these partitions:  441, 4311, 4221, 42111, 411111.
%p A238591 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A238591       `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
%p A238591     end:
%p A238591 a:= n-> add(b(n-5*i, i), i=1..n/5):
%p A238591 seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 03 2014
%t A238591 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 4*Min[p]]], {n, 50}]
%t A238591 (* Second program: *)
%t A238591 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
%t A238591 a[n_] := Sum[b[n - 5*i, i], {i, 1, n/5}];
%t A238591 Array[a, 60] (* _Jean-François Alcover_, Jun 09 2021, after _Alois P. Heinz_ *)
%o A238591 (PARI) my(N=50, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=k, N, 1-x^j)))) \\ _Seiichi Manyama_, May 17 2023
%Y A238591 Cf. A117989, A238589, A238590.
%Y A238591 Cf. A237826, A363067.
%K A238591 nonn,easy
%O A238591 1,7
%A A238591 _Clark Kimberling_, Mar 01 2014