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Number of partitions of n such that the least part occurs exactly three times.
5

%I #11 Mar 01 2014 15:08:56

%S 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 320,405,483,602,726,888,1064,1306,1554,1884,2248,2707,3213,3860,4560,

%U 5446,6435,7638,8990,10651,12494,14734,17260,20277,23683,27754,32328

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

%C 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

%F 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_

%t 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 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Length[p] + 2*Min[p]]], {n, 50}] (* _Clark Kimberling_, Feb 28 2014 *)

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

%K nonn

%O 1,6

%A _Robert G. Wilson v_, Jul 24 2004