%I #6 Feb 07 2014 11:34:15
%S 1,1,2,2,2,3,3,3,3,5,5,5,5,5,7,7,8,6,7,7,11,11,11,11,11,10,11,15,15,
%T 17,15,14,13,15,15,22,22,23,23,21,22,19,20,22,30,30,33,30,33,25,29,25,
%U 29,30,42,42,45,44,43,41,42,36,36,39,42,56,56,62,58,60
%N Triangular array: t(n,k) = number of partitions of n that include a partition of k.
%H Clark Kimberling, <a href="/A235130/b235130.txt">Table of n, a(n) for n = 1..300</a>
%F t(n,1) = A000041(n-1) for n>=0; t(n,n) = A000041(n) for n >= 1.
%e The eleven partitions of 6 include the following six, written as multisets: {1,1,1,1,1,1}, {1,1,1,1,2}, {1,1,2,2}, {1,1,1,3}, {1,2,3}, {3,3}; each has a sub-multiset of which the sum of terms is 3. None of the remaining five partitions of 6 has this property, so t(6,3) = 6. First 7 rows:
%e 1
%e 1 ... 2
%e 2 ... 2 ... 3
%e 3 ... 3 ... 3 ... 5
%e 5 ... 5 ... 5 ... 5 ... 7
%e 7 ... 8 ... 6 ... 7 ... 7 ... 11
%e 11 .. 11 .. 11 .. 11 .. 10 .. 11 .. 15
%t p[n_] := p[n] = IntegerPartitions[n]; t = Table[Length[Cases[p[n], Apply[Alternatives, Map[Flatten[{___, #, ___}] &, p[k]]]]], {n, 15}, {k, n}]; u = Flatten[t] (* 235130 *)
%t TableForm[t] (* _Peter J. C. Moses_, Jan 04 2014 *)
%Y Cf. A000041.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Jan 03 2014
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