A128545 Triangle, read by rows, where T(n,k) is the coefficient of q^(n*k) in the q-binomial coefficient [2*n, n] for n >= k >= 0.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 8, 5, 1, 1, 7, 18, 18, 7, 1, 1, 11, 39, 58, 39, 11, 1, 1, 15, 75, 155, 155, 75, 15, 1, 1, 22, 141, 383, 526, 383, 141, 22, 1, 1, 30, 251, 867, 1555, 1555, 867, 251, 30, 1, 1, 42, 433, 1860, 4192, 5448, 4192, 1860, 433, 42, 1

Offset

0,5

Comments

Variant of A047812 (Parker's partition triangle).

Column 1 equals the number of partitions of n: A000041(n) is the coefficient of q^n in the central q-binomial coefficient [2*n, n] for n > 0.

Links

Paul D. Hanna, Rows n = 0..45, flattened.

Formula

Row sums equal the row sums of triangle A123610: A123611(n) = 2*A047996(2*n,n) = 2*A003239(n) for n > 0, where A047996 is the triangle of circular binomial coefficients and A003239(n) = number of rooted planar trees with n non-root nodes.

Example

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  5,   8,    5,    1;
  1,  7,  18,   18,    7,    1;
  1, 11,  39,   58,   39,   11,    1;
  1, 15,  75,  155,  155,   75,   15,    1;
  1, 22, 141,  383,  526,  383,  141,   22,   1;
  1, 30, 251,  867, 1555, 1555,  867,  251,  30,  1;
  1, 42, 433, 1860, 4192, 5448, 4192, 1860, 433, 42, 1;
  ...

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==0, 1, polcoeff(prod(j=n+1, 2*n, 1-q^j)/prod(j=1, n, 1-q^j), n*k, q)))

Crossrefs

Cf. A003239, A047812 (variant), A047996, A123610, A123611 (row sums).

Cf. A000041 (column 1), A128552 (column 2), A128553 (column 3), A128554 (column 4).

Sequence in context: A181031 A214987 A203946 * A194672 A034364 A183610

Adjacent sequences: A128542 A128543 A128544 * A128546 A128547 A128548

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 10 2007

Status

approved