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Triangle read by rows: T(n,k) is number of partitions of n having at least k distinct parts (n >= 1, k >= 1).
2

%I #39 Apr 01 2017 21:10:48

%S 1,2,3,1,5,2,7,5,11,7,1,15,13,2,22,18,5,30,27,10,42,38,16,1,56,54,27,

%T 2,77,71,42,5,101,99,62,10,135,131,87,20,176,172,128,31,1,231,226,171,

%U 54,2,297,295,236,82,5,385,379,311,127,10,490,488,417,182,20

%N Triangle read by rows: T(n,k) is number of partitions of n having at least k distinct parts (n >= 1, k >= 1).

%C From _Omar E. Pol_, Sep 14 2016: (Start)

%C Row n has length A003056(n) hence the first element of column k is in row A000217(k).

%C Row sums give A000070.

%C Alternating row sums give A090794.

%C Column 1 is A000041, n >= 1. (End)

%C [0, 0] together with column 2 gives A144300. - _Omar E. Pol_, Sep 17 2016

%D Jacques Barbot, Essai sur la structuration de l'analyse combinatoire, Paris, Dulac, 1973, Annexe 12A, p. 74.

%H Alois P. Heinz, <a href="/A261555/b261555.txt">Rows n = 1..500, flattened</a>

%F T(n,k) = Sum_{j>=k} A116608(n,j) assuming A116608(n,j)=0 when j>A003056(n).

%F T(n,1) - T(n,2) = A000005(n). - _Omar E. Pol_, Sep 17 2016

%e Triangle starts:

%e 1;

%e 2;

%e 3, 1;

%e 5, 2;

%e 7, 5;

%e 11, 7, 1;

%e 15, 13, 2;

%e 22, 18, 5;

%e 30, 27, 10;

%e 42, 38, 16, 1;

%e 56, 54, 27, 2;

%e 77, 71, 42, 5;

%e ...

%t Table[DeleteCases[Map[Count[Map[Length@ Union@ # &, IntegerPartitions@ n], k_ /; k >= #] &, Range@ n], 0], {n, 19}] // Flatten (* _Michael De Vlieger_, Sep 14 2016 *)

%Y Cf. A000005, A000041, A000070, A000217, A003056, A060177, A090794, A116608, A144300.

%K nonn,tabf

%O 1,2

%A _Michel Marcus_, Aug 24 2015

%E More terms from _Alois P. Heinz_, Aug 24 2015