A137153 Triangle, read by rows, where T(n,k) = C(2^k + n-k-1, n-k).

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

Offset: 0

Author

Paul D. Hanna, Jan 24 2008

Keywords

• nonn
• tabl

Comments

Matrix inverse is A137156.

T(n,k) is the number of relations between a set of k distinguishable elements and a set of n-k indistinguishable elements. - Isaac R. Browne, Jun 04 2025

Examples

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

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1080; rows 0..45 of flattened triangle.

Crossrefs

Cf. A137154 (row sums), A137155 (antidiagonal sums), A060690 (central terms); A137156 (matrix inverse).

Cf. A092056 (same with reflected rows).

Programs

Mathematica

Table[Binomial[2^k+n-k-1,n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 06 2017 *)

PARI

{T(n,k)=binomial(2^k+n-k-1,n-k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

PARI

{T(n, k) = polcoeff(1/(1-x+x*O(x^n))^(2^k), n-k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))