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

0,8

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.

Links

Table of n, a(n) for n=0..77.

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.

Formula

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

Example

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;
  ...

Crossrefs

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

Sequence in context: A330958 A327186 A021306 * A125300 A371741 A303992

Adjacent sequences: A214278 A214279 A214280 * A214282 A214283 A214284

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Jul 09 2012

Status

approved