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 A235130 Triangular array:  t(n,k) = number of partitions of n that include a partition of k. 4
 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, 17, 15, 14, 13, 15, 15, 22, 22, 23, 23, 21, 22, 19, 20, 22, 30, 30, 33, 30, 33, 25, 29, 25, 29, 30, 42, 42, 45, 44, 43, 41, 42, 36, 36, 39, 42, 56, 56, 62, 58, 60 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Clark Kimberling, Table of n, a(n) for n = 1..300 FORMULA t(n,1) = A000041(n-1) for n>=0; t(n,n) = A000041(n) for n >= 1. EXAMPLE 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: 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 MATHEMATICA 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 *) TableForm[t] (* Peter J. C. Moses, Jan 04 2014 *) CROSSREFS Cf. A000041. Sequence in context: A157873 A022870 A237050 * A131410 A202453 A259529 Adjacent sequences:  A235127 A235128 A235129 * A235131 A235132 A235133 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Jan 03 2014 STATUS approved

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Last modified November 17 22:28 EST 2018. Contains 317279 sequences. (Running on oeis4.)