A121314 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 7, 15, 10, 1, 0, 1, 9, 28, 35, 15, 1, 0, 1, 11, 45, 84, 70, 21, 1, 0, 1, 13, 66, 165, 210, 126, 28, 1, 0, 1, 15, 91, 286, 495, 462, 210, 36, 1

Offset

0,9

Comments

A054142 with first diagonal 1, 0, 0, 0, 0, 0, 0, 0, ...

Mirror image of triangle in A165253.

Links

Alois P. Heinz, Rows n = 0..140, flattened

F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.

Formula

T(0,0)=1; T(n,0)=0 for n > 0; T(n+1,k+1) = binomial(2*n-k,k)for n >= 0 and k >= 0.

Sum_{k=0..n} T(n,k)*x^k = A001519(n), A047849(n), A165310(n), A165311(n), A165312(n), A165314(n), A165322(n), A165323(n), A165324(n) for x = 1,2,3,4,5,6,7,8,9 respectively.

Sum_{k=0..n} 2^k*T(n,k) = (4^n+2)/3.

Sum_{k=0..n} 2^(n-k)*T(n,k) = A001835(n).

Sum_{k=0..n} 3^k*4^(n-k)*T(n,k) = A054879(n). - Philippe Deléham, Aug 26 2006

Sum_{k=0..n} T(n,k)*(-1)^k*2^(3n-2k) = A143126(n). - Philippe Deléham, Oct 31 2008

Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A138340(n)/4^n. - Philippe Deléham, Nov 01 2008

G.f.: (1-(y+1)*x)/(1-(2y+1)*x+y^2*x^2). - Philippe Deléham, Nov 01 2011

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0. - Philippe Deléham, Feb 19 2012

Example

Triangle begins
  1;
  0,  1;
  0,  1,  1;
  0,  1,  3,  1;
  0,  1,  5,  6,  1;
  0,  1,  7, 15, 10,  1;
  0,  1,  9, 28, 35, 15,  1;
  0,  1, 11, 45, 84, 70, 21,  1;

Crossrefs

Cf. A054142, A165253.

Sequence in context: A354821 A081719 A327618 * A119271 A364310 A323222

Adjacent sequences: A121311 A121312 A121313 * A121315 A121316 A121317

Keyword

nonn,tabl

Author

Philippe Deléham, Aug 25 2006

Status

approved