This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A097093 #19 Jul 05 2025 04:53:59
%S A097093 0,0,0,0,1,0,1,1,2,3,4,4,8,9,14,16,23,27,39,48,62,76,100,120,159,190,
%T A097093 241,292,367,443,552,663,816,980,1200,1430,1742,2075,2504,2979,3575,
%U A097093 4232,5063,5980,7114,8382,9930,11663,13773,16140,18980,22190,26017
%N A097093 Number of partitions of n such that the least part occurs exactly five times.
%C A097093 In general, if k>=1 and g.f. = Sum_{m>0} (x^(k*m) / Product_{i>m} (1-x^i)), then a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + (24*k - 35)*Pi/(24*sqrt(6)))/sqrt(n)). - _Vaclav Kotesovec_, Jul 05 2025
%H A097093 Vaclav Kotesovec, <a href="/A097093/b097093.txt">Table of n, a(n) for n = 1..10000</a>
%F A097093 G.f.: Sum_{m>0} (x^(5*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 A097093 From _Vaclav Kotesovec_, Jul 05 2025: (Start)
%F A097093 a(n) = -p(n) + 5*p(n+5) - p(n+6) - 2*p(n+7) - 2*p(n+8) - 3*p(n+9) + 3*p(n+10) + p(n+11) + 2*p(n+12) - p(n+13) - 2*p(n+14) + p(n+15), where p(n) = A000041(n).
%F A097093 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + 85*Pi/(24*sqrt(6)))/sqrt(n)). (End)
%t A097093 f[n_] := Block[{p = IntegerPartitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 6]; If[ q[[1]] != q[[6]] && q[[2]] == q[[6]], c++ ]; k++ ]; c]; Table[ f[n], {n, 54}]
%t A097093 Table[Count[IntegerPartitions[n],_?(Length[Split[#][[-1]]]==5&)],{n,60}] (* _Harvey P. Dale_, Feb 07 2022 *)
%t A097093 nmax = 60; Rest[CoefficientList[Series[Sum[x^(5*m)/Product[1-x^k,{k,m+1,nmax}], {m, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jul 04 2025 *)
%t A097093 Table[-PartitionsP[n] + 5 PartitionsP[5 + n] - PartitionsP[6 + n] - 2 PartitionsP[7 + n] - 2 PartitionsP[8 + n] - 3 PartitionsP[9 + n] + 3 PartitionsP[10 + n] + PartitionsP[11 + n] + 2 PartitionsP[12 + n] - PartitionsP[13 + n] - 2 PartitionsP[14 + n] + PartitionsP[15 + n], {n, 1, 60}] (* _Vaclav Kotesovec_, Jul 05 2025 *)
%Y A097093 Cf. A002865, A096373, A097091, A097092.
%K A097093 nonn
%O A097093 1,9
%A A097093 _Robert G. Wilson v_, Jul 24 2004