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A114638 Number of partitions of n such that number of parts is equal to the sum of parts counted without multiplicities.

%I A114638 #28 Oct 27 2019 20:58:25
%S A114638 1,1,0,0,2,1,1,0,2,2,3,5,5,6,9,7,8,14,12,16,21,28,32,43,47,61,68,84,
%T A114638 89,109,126,140,170,198,227,261,323,362,427,501,581,658,794,880,1036,
%U A114638 1175,1355,1526,1776,1985,2281,2588,2943,3312,3799,4271,4852,5497
%N A114638 Number of partitions of n such that number of parts is equal to the sum of parts counted without multiplicities.
%C A114638 The Heinz numbers of these integer partitions are given by A324570. - _Gus Wiseman_, Mar 09 2019
%H A114638 Alois P. Heinz, <a href="/A114638/b114638.txt">Table of n, a(n) for n = 0..500</a>
%e A114638 a(10) = 3 because we have [5,1,1,1,1,1], [3,3,3,1] and [3,2,2,1,1,1].
%e A114638 From _Gus Wiseman_, Mar 09 2019: (Start)
%e A114638 The a(1) = 1 through a(12) = 5 integer partitions (empty columns not shown):
%e A114638   1  22   221  3111  3311   333     3331    32222    33222
%e A114638      211             41111  321111  322111  44111    322221
%e A114638                                     511111  322211   332211
%e A114638                                             332111   4221111
%e A114638                                             4211111  6111111
%e A114638 (End)
%p A114638 a:=proc(n) local P,c,j,S: with(combinat): P:=partition(n): c:=0: for j from 1 to nops(P) do S:=convert(P[j],set): if nops(P[j])=sum(S[i],i=1..nops(S)) then c:=c+1 else c:=c fi: c: od: end: seq(a(n), n=0..35); # _Emeric Deutsch_, Mar 01 2006
%t A114638 a[n_] := Module[{P, c, j, S}, P = IntegerPartitions[n]; c = 0; For[j = 1, j <= Length[P], j++, S = Union[P[[j]]]; If[Length[P[[j]]] == Total[S],  c++] ]; c];
%t A114638 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 07 2018, after _Emeric Deutsch_ *)
%o A114638 (PARI) apply( A114638(n,s=0)={forpart(p=n,#p==vecsum(Set(p))&&s++); s}, [0..50]) \\ _M. F. Hasler_, Oct 27 2019
%Y A114638 Cf. A003114, A006141, A039900, A047993, A064174, A066328, A243149 (the same for compositions).
%Y A114638 Cf. A324516, A324518, A324520, A324521, A324570.
%Y A114638 Cf. A116861 (number of partitions of n having a given sum of distinct parts).
%K A114638 nonn
%O A114638 0,5
%A A114638 _Vladeta Jovovic_, Feb 18 2006
%E A114638 More terms from _Emeric Deutsch_, Mar 01 2006