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A261555 Triangle read by rows: T(n,k) is number of partitions of n having at least k distinct parts (n >= 1, k >= 1).

%I A261555 #39 Apr 01 2017 21:10:48
%S A261555 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 A261555 2,77,71,42,5,101,99,62,10,135,131,87,20,176,172,128,31,1,231,226,171,
%U A261555 54,2,297,295,236,82,5,385,379,311,127,10,490,488,417,182,20
%N A261555 Triangle read by rows: T(n,k) is number of partitions of n having at least k distinct parts (n >= 1, k >= 1).
%C A261555 From _Omar E. Pol_, Sep 14 2016: (Start)
%C A261555 Row n has length A003056(n) hence the first element of column k is in row A000217(k).
%C A261555 Row sums give A000070.
%C A261555 Alternating row sums give A090794.
%C A261555 Column 1 is A000041, n >= 1. (End)
%C A261555 [0, 0] together with column 2 gives A144300. - _Omar E. Pol_, Sep 17 2016
%D A261555 Jacques Barbot, Essai sur la structuration de l'analyse combinatoire, Paris, Dulac, 1973, Annexe 12A, p. 74.
%H A261555 Alois P. Heinz, <a href="/A261555/b261555.txt">Rows n = 1..500, flattened</a>
%F A261555 T(n,k) = Sum_{j>=k} A116608(n,j) assuming A116608(n,j)=0 when j>A003056(n).
%F A261555 T(n,1) - T(n,2) = A000005(n). - _Omar E. Pol_, Sep 17 2016
%e A261555 Triangle starts:
%e A261555 1;
%e A261555 2;
%e A261555 3,  1;
%e A261555 5,  2;
%e A261555 7,  5;
%e A261555 11, 7,  1;
%e A261555 15, 13, 2;
%e A261555 22, 18, 5;
%e A261555 30, 27, 10;
%e A261555 42, 38, 16, 1;
%e A261555 56, 54, 27, 2;
%e A261555 77, 71, 42, 5;
%e A261555 ...
%t A261555 Table[DeleteCases[Map[Count[Map[Length@ Union@ # &, IntegerPartitions@ n], k_ /; k >= #] &, Range@ n], 0], {n, 19}] // Flatten (* _Michael De Vlieger_, Sep 14 2016 *)
%Y A261555 Cf. A000005, A000041, A000070, A000217, A003056, A060177, A090794, A116608, A144300.
%K A261555 nonn,tabf
%O A261555 1,2
%A A261555 _Michel Marcus_, Aug 24 2015
%E A261555 More terms from _Alois P. Heinz_, Aug 24 2015