A092056 Square table read by downward antidiagonals where T(n,k) = binomial(n+2^k-1,n).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 10, 4, 1, 1, 16, 36, 20, 5, 1, 1, 32, 136, 120, 35, 6, 1, 1, 64, 528, 816, 330, 56, 7, 1, 1, 128, 2080, 5984, 3876, 792, 84, 8, 1, 1, 256, 8256, 45760, 52360, 15504, 1716, 120, 9, 1, 1, 512, 32896, 357760, 766480, 376992, 54264, 3432, 165, 10, 1

Offset: 0

Author

Henry Bottomley, Feb 19 2004

Keywords

• nonn
• tabl

Comments

Each column is convolution of preceding column starting from the all 1's sequence.

T(n,k) is the number of relations between a set of k distinguishable elements and a set of n indistinguishable elements. - Isaac R. Browne, May 14 2025

Examples

			Rows start:
  1, 1,  1,   1,    1,     1,      1,...
  1, 2,  4,   8,   16,    32,     64,...
  1, 3, 10,  36,  136,   528,   2080,...
  1, 4, 20, 120,  816,  5984,  45760,...
  1, 5, 35, 330, 3876, 52360, 766480,...
  ...
		

Crossrefs

Columns include (essentially) A000012, A000027, A000292, A000580, A010968, etc.

Rows include A000012, A000079, A007582, A092056.

Main diagonal gives A060690.

Cf. A137153 (same with reflected antidiagonals).

Formula

T(n,k) = Sum_{i=0..n} T(i,k-1)*T(n-i,k-1) starting with T(n,0) = 1 for n>=0.