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A305556 Irregular triangle read by rows: T(n,k) is the number of superdiagonal bargraphs with area n and with k columns.

%I A305556 #12 Jun 21 2018 02:23:40
%S A305556 1,1,1,1,1,2,1,3,1,4,1,1,5,3,1,6,6,1,7,10,1,8,15,1,1,9,21,4,1,10,28,
%T A305556 10,1,11,36,20,1,12,45,35,1,13,55,56,1,1,14,66,84,5,1,15,78,120,15,1,
%U A305556 16,91,165,35,1,17,105,220,70,1,18,120,286,126,1,19,136,364,210,1,1,20,153,455
%N A305556 Irregular triangle read by rows: T(n,k) is the number of superdiagonal bargraphs with area n and with k columns.
%H A305556 Emeric Deutsch, Emanuele Munarini, Simone Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2009.12.013">Skew Dyck paths, area, and superdiagonal bargraphs</a>, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.
%F A305556 T(n,k) = binomial(n-1-k*(k-1)/2,k-1), 1<=k <= (sqrt(1+8*n)-1)/2.
%e A305556    1,
%e A305556    1,
%e A305556    1,   1,
%e A305556    1,   2,
%e A305556    1,   3,
%e A305556    1,   4,   1,
%e A305556    1,   5,   3,
%e A305556    1,   6,   6,
%e A305556    1,   7,  10,
%e A305556    1,   8,  15,   1,
%e A305556    1,   9,  21,   4,
%e A305556    1,  10,  28,  10,
%e A305556    1,  11,  36,  20,
%e A305556    1,  12,  45,  35,
%e A305556    1,  13,  55,  56,   1,
%e A305556    1,  14,  66,  84,   5,
%e A305556    1,  15,  78, 120,  15,
%e A305556    1,  16,  91, 165,  35,
%e A305556    1,  17, 105, 220,  70,
%e A305556    1,  18, 120, 286, 126,
%e A305556    1,  19, 136, 364, 210,   1,
%e A305556    1,  20, 153, 455, 330,   6,
%e A305556    1,  21, 171, 560, 495,  21,
%e A305556    1,  22, 190, 680, 715,  56,
%e A305556    1,  23, 210, 816,1001, 126,
%e A305556    1,  24, 231, 969,1365, 252,
%e A305556    1,  25, 253,1140,1820, 462,
%e A305556    1,  26, 276,1330,2380, 792,   1,
%e A305556    1,  27, 300,1540,3060,1287,   7,
%e A305556    1,  28, 325,1771,3876,2002,  28,
%p A305556 A305556 := proc(n,k)
%p A305556     binomial(n-binomial(k,2)-1,k-1) ;
%p A305556 end proc:
%p A305556 for n from 0 to 30 do
%p A305556 for k from 1 to floor((sqrt(1+8*n)-1)/2) do
%p A305556     printf("%d,",A305556(n,k)) ;
%p A305556 end do:
%p A305556 end do:
%Y A305556 Cf. A219282 (row sums), A000217 (column 3), A000292 (column 4), A000332 (column 5)
%K A305556 nonn,tabf,easy
%O A305556 1,6
%A A305556 _R. J. Mathar_, Jun 21 2018