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A136406
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Triangle read by rows: T(n,k) is the number of bi-partitions of the pair (n,k) into pairs (n_i,k_i) of positive integers such that sum k_i = k and sum n_i*k_i^2 = n.
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4
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1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 1, 3, 4, 3, 1, 1, 1, 3, 5, 4, 3, 1, 1, 1, 5, 6, 8, 4, 3, 1, 1, 1, 4, 10, 8, 8, 4, 3, 1, 1, 1, 5, 10, 14, 11, 8, 4, 3, 1, 1, 1, 5, 12, 16, 17, 11, 8, 4, 3, 1, 1, 1, 7, 14, 23, 21, 21, 11, 8, 4, 3, 1, 1, 1, 6, 17, 25, 32, 24, 21, 11, 8, 4, 3, 1, 1
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OFFSET
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1,8
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COMMENTS
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T(n,1) = T(n,n) = 1.
T(n,n-k) does not depend on k if k <= floor(n/2).
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LINKS
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EXAMPLE
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Triangle begins:
1,
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 2, 3, 1, 1;
1, 3, 4, 3, 1, 1;
1, 3, 5, 4, 3, 1, 1;
1, 5, 6, 8, 4, 3, 1, 1;
1, 4, 10, 8, 8, 4, 3, 1, 1;
1, 5, 10, 14, 11, 8, 4, 3, 1, 1;
1, 5, 12, 16, 17, 11, 8, 4, 3, 1, 1;
...
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PROG
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(PARI)
P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}
T(n)={Vecrev(polcoef(prod(w=1, sqrtint(n), sum(k=0, n\w^2, (x^w*y)^(k*w) / P(k, w^2, n))), n)/y)}
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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