A059328 Table T(n,k) = T(n - 1,k) + T(n,k - 1) + T(n - 1,k)*T(n,k - 1) starting with T(0,0)=1, read by antidiagonals.

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 63, 15, 1, 1, 31, 1023, 1023, 31, 1, 1, 63, 32767, 1048575, 32767, 63, 1, 1, 127, 2097151, 34359738367, 34359738367, 2097151, 127, 1, 1, 255, 268435455, 72057594037927935, 1180591620717411303423, 72057594037927935, 268435455, 255, 1

Offset: 0

Author

Henry Bottomley, Jan 26 2001

Keywords

• nonn
• tabl

Comments

In binary representation T(n,k) is the concatenation of T(n-1,k-1) and T(n-1,k), 0Reinhard Zumkeller, Jan 23 2003

Examples

			Triangle T(n,k) begins:
  1;
  1,  1;
  1,  3,     1;
  1,  7,     7,       1;
  1, 15,    63,      15,     1;
  1, 31,  1023,    1023,    31,  1;
  1, 63, 32767, 1048575, 32767, 63, 1;
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 14 rows, flattened

Crossrefs

Columns k=0-2 give: A000012, A000225, A126883(n-1).

Row sums give A306020.

Cf. A007318.

Programs

Mathematica

Table[2^(Binomial[n, k]) - 1, {n, 0, 5}, {k, 0, n}] (* G. C. Greubel, Jan 07 2017 *)

Python

from math import comb, isqrt
def A059328(n): return (1<Chai Wah Wu, Apr 30 2025

Formula

T(n, k) = 2^C(n+k, n)-1; a(n) = 2^A007318(n)-1.

If U(n, k) := 1 + T(n, k), then U(n, k) = U(n-1, k) * U(n, k-1). - Michael Somos, Jan 07 2017