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A128545
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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.
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7
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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
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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;
...
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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