A214281 Triangle by rows, row n contains the ConvOffs transform of the first n terms of 1, 1, 3, 2, 5, 3, 7, ... (A026741 without leading zero).

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 6, 2, 1, 1, 5, 10, 10, 5, 1, 1, 3, 15, 10, 15, 3, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 4, 28, 28, 70, 28, 28, 4, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1, 5, 45, 60, 210, 126, 210, 60, 45, 5, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Author

Gary W. Adamson, Jul 09 2012

Keywords

• nonn
• tabl
• easy

Comments

The ConvOffs transform of a sequence s(0), s(1), ..., s(t-1) is defined by a(0)=1 and a(n) = a(n-1)*s(t-n)/s(n-1) for 1 <= n < t. An example of this process is also shown in the Narayana triangle, A001263. By increasing the length t of the input sequence (here: A026741) we create more and more rows of the triangle.

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  3,   1;
  1,  2,  6,   2,   1;
  1,  5, 10,  10,   5,   1;
  1,  3, 15,  10,  15,   3,   1;
  1,  7, 21,  35,  35,  21,   7,   1;
  1,  4, 28,  28,  70,  28,  28,   4,   1;
  1,  9, 36,  84, 126, 126,  84,  36,   9,  1;
  1,  5, 45,  60, 210, 126, 210,  60,  45,  5,  1;
  1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1;
  ...
		

Links

• Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Crossrefs

Cf. A134683 (row sums), A026741, A001263.

Formula

T(n,k) = binomial(n,k) if n is odd.