A175331 Array A092921(n,k) without the first two rows, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 2, 1, 1, 8, 7, 4, 2, 1, 1, 13, 13, 8, 4, 2, 1, 1, 21, 24, 15, 8, 4, 2, 1, 1, 34, 44, 29, 16, 8, 4, 2, 1, 1, 55, 81, 56, 31, 16, 8, 4, 2, 1, 1, 89, 149, 108, 61, 32, 16, 8, 4, 2, 1, 1, 144, 274, 208, 120, 63, 32, 16, 8, 4, 2, 1, 1, 233, 504, 401, 236, 125, 64, 32, 16, 8, 4, 2, 1

Offset

2,5

Comments

Antidiagonal sums are A048888. This is a transposed version of A048887, so the bivariate generating function is obtained by swapping the two arguments.

Brlek et al. (2006) call this table "number of psp-polyominoes with flat bottom". - N. J. A. Sloane, Oct 30 2018

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 125, 155.

Links

Table of n, a(n) for n=2..92.

Srecko Brlek, Andrea Frosini, Simone Rinaldi, Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol.13, (2006). Table 1 is essentially this array. - N. J. A. Sloane, Jul 20 2014

Formula

T(n,k) = A092921(n,k), n >= 2.

T(n,2) = A000045(n).

T(n,3) = A000073(n+2).

T(n,4) = A000078(n+2).

Example

The array starts in row n=2 with columns k >= 1 as:
  1   1   1   1   1   1   1   1   1   1
  1   2   2   2   2   2   2   2   2   2
  1   3   4   4   4   4   4   4   4   4
  1   5   7   8   8   8   8   8   8   8
  1   8  13  15  16  16  16  16  16  16
  1  13  24  29  31  32  32  32  32  32
  1  21  44  56  61  63  64  64  64  64
  1  34  81 108 120 125 127 128 128 128
  1  55 149 208 236 248 253 255 256 256

Maple

A092921 := proc(n, k) if k <= 0 or n <= 0 then 0; elif k = 1 or n = 1 then 1; else add( procname(n-i, k), i=1..k) ; end if; end proc:
A175331 := proc(n, k) A092921(n, k) ; end proc: # R. J. Mathar, Dec 17 2010

Mathematica

f[x_, n_] = (x - x^(m + 1))/(1 - 2*x + x^(m + 1))
a = Table[Table[SeriesCoefficient[
      Series[f[x, m], {x, 0, 10}], n], {n, 0, 10}], {m, 1, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}];
Flatten[%]

Crossrefs

Cf. A000045, A048887, A048004, A126198.

Sequence in context: A152462 A180360 A318805 * A098805 A168396 A049286

Adjacent sequences: A175328 A175329 A175330 * A175332 A175333 A175334

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Dec 03 2010

Status

approved