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A128562
Triangle, read by rows, where T(n,k) is the coefficient of q^((n+1)*k) in the q-binomial coefficient [2*n+1, n] for n >= k >= 0.
1
1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 12, 6, 1, 1, 10, 29, 29, 10, 1, 1, 14, 61, 94, 61, 14, 1, 1, 21, 120, 263, 263, 120, 21, 1, 1, 29, 222, 645, 910, 645, 222, 29, 1, 1, 41, 392, 1468, 2724, 2724, 1468, 392, 41, 1, 1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1
OFFSET
0,5
COMMENTS
Row sums equal a shifted version of A003239 (number of rooted planar trees with n non-root nodes). Column 1 is a shifted version of A000065 (-1 + number of partitions of n). Column 2 is a shifted version of A128563. This array is a variant of triangles A128545 and A047812 (Parker's partition triangle).
FORMULA
T(n,k) = [q^((n+1)*k)] Product_{j=n+1..2*n+1}(1-q^j) / Product_{j=1..n+1}(1-q^j).
EXAMPLE
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 6, 12, 6, 1;
1, 10, 29, 29, 10, 1;
1, 14, 61, 94, 61, 14, 1;
1, 21, 120, 263, 263, 120, 21, 1;
1, 29, 222, 645, 910, 645, 222, 29, 1;
1, 41, 392, 1468, 2724, 2724, 1468, 392, 41, 1;
1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1;
...
PROG
(PARI) T(n, k)=if(n<k || k<0, 0, if(n==0, 1, polcoeff(prod(j=n+1, 2*n+1, 1-q^j)/prod(j=1, n+1, 1-q^j), (n+1)*k, q)))
CROSSREFS
Cf. A000065 (column 1), A003239 (row sums), A128563 (column 2).
Variants are A047812 and A128545.
Sequence in context: A096806 A116672 A161126 * A034368 A361745 A296157
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 10 2007
EXTENSIONS
Minor edits by Petros Hadjicostas, Jun 01 2020
STATUS
approved