login
Number of partitions of n such that the least part occurs exactly four times.
4

%I #13 Jan 18 2021 13:46:46

%S 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 253,303,385,459,572,687,852,1014,1244,1482,1807,2145,2595,3075,3701,

%U 4375,5231,6170,7350,8641,10247,12025,14201,16620,19557,22839,26790,31209

%N Number of partitions of n such that the least part occurs exactly four times.

%C 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

%F 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_

%t 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 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Length[p] + 3*Min[p]]], {n, 50}] (* _Clark Kimberling_, Feb 28 2014 *)

%t Table[Count[IntegerPartitions[n],_?(Length[Split[#][[-1]]]==4&)],{n,60}] (* _Harvey P. Dale_, Jan 18 2021 *)

%Y Cf. A002865, A096373, A097091, A097093.

%K nonn

%O 1,8

%A _Robert G. Wilson v_, Jul 24 2004