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A097091 Number of partitions of n such that the least part occurs exactly three times.

%I A097091 #20 Jul 05 2025 04:53:18
%S A097091 0,0,1,0,1,2,2,2,6,5,8,11,15,18,27,30,43,54,69,83,113,134,172,211,265,
%T A097091 320,405,483,602,726,888,1064,1306,1554,1884,2248,2707,3213,3860,4560,
%U A097091 5446,6435,7638,8990,10651,12494,14734,17260,20277,23683,27754,32328
%N A097091 Number of partitions of n such that the least part occurs exactly three times.
%C A097091 Number of partitions p of n such that 2*min(p) + (number of parts of p) is a part of p. - _Clark Kimberling_, Feb 28 2014
%H A097091 Vaclav Kotesovec, <a href="/A097091/b097091.txt">Table of n, a(n) for n = 1..10000</a>
%F A097091 G.f.: Sum_{m>0} (x^(3*m) / Product_{i>m} (1-x^i)). More generally, g.f. for number of partitions of n such that the least part occurs exactly k times is Sum_{m>0} (x^(k*m) / Product_{i>m} (1-x^i)). _Vladeta Jovovic_
%F A097091 From _Vaclav Kotesovec_, Jul 05 2025: (Start)
%F A097091 a(n) = -p(n) + 3*p(n+3) - p(n+4) - 2*p(n+5) + p(n+6), where p(n) = A000041(n).
%F A097091 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + 37*Pi/(24*sqrt(6)))/sqrt(n)). (End)
%t A097091 a[n_] := Module[{p = IntegerPartitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[p[[k]], 4]; If[q[[1]] != q[[4]] && q[[2]] == q[[4]], c++]; k++]; c]; Table[ a[n], {n, 52}]
%t A097091 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Length[p] + 2*Min[p]]], {n, 50}] (* _Clark Kimberling_, Feb 28 2014 *)
%t A097091 nmax = 60; Rest[CoefficientList[Series[Sum[x^(3*m)/Product[1-x^k,{k,m+1,nmax}], {m, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jul 04 2025 *)
%t A097091 Table[-PartitionsP[n] + 3 PartitionsP[3 + n] - PartitionsP[4 + n] - 2 PartitionsP[5 + n] + PartitionsP[6 + n], {n, 1, 60}] (* _Vaclav Kotesovec_, Jul 05 2025 *)
%Y A097091 Cf. A002865, A096373, A097092, A097093.
%K A097091 nonn
%O A097091 1,6
%A A097091 _Robert G. Wilson v_, Jul 24 2004