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

0,5

Comments

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

Links

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

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-1, k-1). - Michael Somos, Jan 07 2017

Example

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;
  ...

Mathematica

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

Prog

(Python)
from math import comb, isqrt
def A059328(n): return (1<<comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)), n-comb(r+1, 2)))-1 # Chai Wah Wu, Apr 30 2025

Crossrefs

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

Row sums give A306020.

Cf. A007318.

Sequence in context: A141689 A058669 A057004 * A174387 A176791 A259471

Adjacent sequences: A059325 A059326 A059327 * A059329 A059330 A059331

Keyword

nonn,tabl

Author

Henry Bottomley, Jan 26 2001

Status

approved