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

1,6

Links

Table of n, a(n) for n=1..80.

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.

Formula

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

Example

   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,

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:

Crossrefs

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

Sequence in context: A360285 A226248 A102547 * A177812 A280504 A087267

Adjacent sequences: A305553 A305554 A305555 * A305557 A305558 A305559

Keyword

nonn,tabf,easy

Author

R. J. Mathar, Jun 21 2018

Status

approved