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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 [2n,n] for n>=k>=0.
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4
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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 [2n,n] for n>0.
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FORMULA
| Row sums equal the row sums of triangle A123610: A123611(n) = 2*A047996(2n,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
| Triangle 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
| (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
| Cf. A123611 (row sums), A047812 (variant).
Cf. A000041 (column 1), A128552 (column 2), A128553(column 3), A128554 (column 4).
Sequence in context: A089980 A181031 A203946 * A194672 A034364 A090011
Adjacent sequences: A128542 A128543 A128544 * A128546 A128547 A128548
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Mar 10 2007
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