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

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, 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, 16, 91, 165, 35, 1, 17, 105, 220, 70, 1, 18, 120, 286, 126, 1, 19, 136, 364, 210, 1, 1, 20, 153, 455

Offset: 1

Author

R. J. Mathar, Jun 21 2018

Keywords

• nonn
• tabf
• easy

Examples

			   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,  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,  16,  91, 165,  35,
   1,  17, 105, 220,  70,
   1,  18, 120, 286, 126,
   1,  19, 136, 364, 210,   1,
   1,  20, 153, 455, 330,   6,
   1,  21, 171, 560, 495,  21,
   1,  22, 190, 680, 715,  56,
   1,  23, 210, 816,1001, 126,
   1,  24, 231, 969,1365, 252,
   1,  25, 253,1140,1820, 462,
   1,  26, 276,1330,2380, 792,   1,
   1,  27, 300,1540,3060,1287,   7,
   1,  28, 325,1771,3876,2002,  28,
		

Links

• Emeric Deutsch, Emanuele Munarini, Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.

Crossrefs

Cf. A219282 (row sums), A000217 (column 3), A000292 (column 4), A000332 (column 5)

Programs

Maple

A305556 := proc(n,k)
    binomial(n-binomial(k,2)-1,k-1) ;
end proc:
for n from 0 to 30 do
for k from 1 to floor((sqrt(1+8*n)-1)/2) do
    printf("%d,",A305556(n,k)) ;
end do:
end do:

Formula

T(n,k) = binomial(n-1-k*(k-1)/2,k-1), 1<=k <= (sqrt(1+8*n)-1)/2.