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

%I #10 Feb 07 2022 12:45:26

%S 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 241,292,367,443,552,663,816,980,1200,1430,1742,2075,2504,2979,3575,

%U 4232,5063,5980,7114,8382,9930,11663,13773,16140,18980,22190,26017

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

%F 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_

%t 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 Table[Count[IntegerPartitions[n],_?(Length[Split[#][[-1]]]==5&)],{n,60}] (* _Harvey P. Dale_, Feb 07 2022 *)

%Y Cf. A002865, A096373, A097091, A097092.

%K nonn

%O 1,9

%A _Robert G. Wilson v_, Jul 24 2004