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A237826 Number of partitions of n such that 4*(least part) = greatest part.

%I A237826 #23 Jun 19 2025 04:04:34
%S A237826 0,0,0,0,1,1,2,3,5,7,9,12,16,20,26,31,38,47,55,67,78,92,106,126,145,
%T A237826 167,190,219,247,288,320,366,410,466,520,591,654,739,820,924,1018,
%U A237826 1148,1263,1415,1562,1740,1911,2136,2342,2607,2859,3169,3469,3849,4208
%N A237826 Number of partitions of n such that 4*(least part) = greatest part.
%H A237826 David A. Corneth, <a href="/A237826/b237826.txt">Table of n, a(n) for n = 1..10000</a>
%F A237826 G.f.: Sum_{k>=1} x^(5*k)/Product_{j=k..4*k} (1-x^j). - _Seiichi Manyama_, May 14 2023
%F A237826 a(n) ~ c * d^sqrt(n) / sqrt(n), where d = 4.9219345... and c = 0.1699648... - _Vaclav Kotesovec_, Jun 19 2025
%e A237826 a(8) = 3 counts these partitions:  431, 4211, 41111.
%t A237826 z = 64; q[n_] := q[n] = IntegerPartitions[n];
%t A237826 Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}]     (* A237825*)
%t A237826 Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}]     (* A237826 *)
%t A237826 Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}]     (* A237827 *)
%t A237826 Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)
%t A237826 Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)
%t A237826 Table[Count[IntegerPartitions[n],_?(#[[1]]==4#[[-1]]&)],{n,60}] (* _Harvey P. Dale_, Jun 15 2023 *)
%t A237826 kmax = 55;
%t A237826 Sum[x^(5k)/Product[1 - x^j, {j, k, 4 k}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* _Jean-François Alcover_, May 30 2024, after _Seiichi Manyama_ *)
%o A237826 (PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=k, 4*k, 1-x^j)))) \\ _Seiichi Manyama_, May 14 2023
%Y A237826 Cf. A000041, A117086, A237757, A237828, A237829.
%Y A237826 Cf. A118096, A237825, A237827.
%K A237826 nonn,easy
%O A237826 1,7
%A A237826 _Clark Kimberling_, Feb 16 2014