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 A097092 #20 Jul 05 2025 04:53:35
%S A097092 0,0,0,1,0,1,1,3,2,4,5,9,9,14,16,26,29,40,48,67,79,105,126,165,196,
%T A097092 253,303,385,459,572,687,852,1014,1244,1482,1807,2145,2595,3075,3701,
%U A097092 4375,5231,6170,7350,8641,10247,12025,14201,16620,19557,22839,26790,31209
%N A097092 Number of partitions of n such that the least part occurs exactly four times.
%C A097092 Number of partitions p of n such that 3*min(p) + (number of parts of p) is a part of p. - _Clark Kimberling_, Feb 28 2014
%H A097092 Vaclav Kotesovec, <a href="/A097092/b097092.txt">Table of n, a(n) for n = 1..10000</a>
%F A097092 G.f.: Sum_{m>0} (x^(4*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 A097092 From _Vaclav Kotesovec_, Jul 05 2025: (Start)
%F A097092 a(n) = -p(n) + 4*p(n+4) - p(n+5) - 2*p(n+6) - 2*p(n+7) + p(n+8) + 2*p(n+9) - p(n+10), where p(n) = A000041(n).
%F A097092 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + 61*Pi/(24*sqrt(6)))/sqrt(n)). (End)
%t A097092 a[n_] := Module[{p = IntegerPartitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 5]; If[ q[[1]] != q[[5]] && q[[2]] == q[[5]], c++ ]; k++ ]; c]; Table[ a[n], {n, 53}]
%t A097092 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Length[p] + 3*Min[p]]], {n, 50}] (* _Clark Kimberling_, Feb 28 2014 *)
%t A097092 Table[Count[IntegerPartitions[n],_?(Length[Split[#][[-1]]]==4&)],{n,60}] (* _Harvey P. Dale_, Jan 18 2021 *)
%t A097092 nmax = 60; Rest[CoefficientList[Series[Sum[x^(4*m)/Product[1-x^k,{k,m+1,nmax}], {m, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jul 04 2025 *)
%t A097092 Table[-PartitionsP[n] + 4 PartitionsP[4 + n] - PartitionsP[5 + n] - 2 PartitionsP[6 + n] - 2 PartitionsP[7 + n] + PartitionsP[8 + n] + 2 PartitionsP[9 + n] - PartitionsP[10 + n], {n, 1, 60}] (* _Vaclav Kotesovec_, Jul 05 2025 *)
%Y A097092 Cf. A002865, A096373, A097091, A097093.
%K A097092 nonn
%O A097092 1,8
%A A097092 _Robert G. Wilson v_, Jul 24 2004