A091913 Triangle read by rows: a(n,k) = C(n,k)*(2^(n-k) - 1) for k= n, where k=0..max(n-1,0).

0, 1, 3, 2, 7, 9, 3, 15, 28, 18, 4, 31, 75, 70, 30, 5, 63, 186, 225, 140, 45, 6, 127, 441, 651, 525, 245, 63, 7, 255, 1016, 1764, 1736, 1050, 392, 84, 8, 511, 2295, 4572, 5292, 3906, 1890, 588, 108, 9, 1023, 5110, 11475, 15240, 13230, 7812, 3150, 840, 135, 10, 2047

Offset: 0

Author

Ross La Haye, Mar 10 2004

Keywords

• easy
• nonn
• tabf

Comments

Row lengths are 1,1,2,3,4,... = A028310. - M. F. Hasler, Jul 21 2012

Rows: Sum of the n-th row = A001047(n); Sum of the n-th row excluding column 0 = A028243(n+1). Columns: a(n,0) = A000225(n); a(n,1) = A058877(n). Diagonals: a(n,n-2) = A045943(n-1). Also note that the sums of the antidiagonals = A006684.

As an infinite lower triangular matrix * the Bernoulli numbers as a vector (Cf. A027641) = the natural numbers: [1, 2, 3, ...]. The same matrix * the Bernoulli number version starting [1, 1/2, 1/6, ...] = A001787: (1, 4, 12, 32, ...). - Gary W. Adamson, Mar 13 2012

Examples

			Triangle begins
   0;
   1;
   3,   2;
   7,   9,   3;
  15,  28,  18,   4;
  31,  75,  70,  30,   5;
  63, 186, 225, 140,  45,   6;
  ...
a(5,3) = 30 because C(5,3) = 10, 2^(5 - 3) - 1 = 3 and 10 * 3 = 30.
		

Crossrefs

Cf. A007318, A038207.

Cf. A001787, A027641, A027642.

Formula

For k>=n, a(n, k) = 0; for k < n, a(n, k) = C(n, k) * (2^(n-k) - 1) = Sum [C(n,k) * C(n-k, m), {m=1 to n-k}]. [Formula corrected Aug 22 2006]

The triangle (1; 3,2; 7,9,3; ...) = A007318^2 - A007318, then delete the right border of zeros. - Gary W. Adamson, Nov 16 2007

O.g.f.: 1/( (1 - (1 + x)*t)*(1 - (2 + x)*t) ) = 1 + (3 + 2*x)*t + (7 + 9*x + 3*x^2)*t^2 + .... - Peter Bala, Jul 16 2013

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004