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A355395
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(j*(n-j)) * binomial(n,j).
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1
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1, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 6, 8, 2, 1, 2, 8, 26, 16, 2, 1, 2, 10, 56, 162, 32, 2, 1, 2, 12, 98, 704, 1442, 64, 2, 1, 2, 14, 152, 2050, 15392, 18306, 128, 2, 1, 2, 16, 218, 4752, 84482, 593408, 330626, 256, 2, 1, 2, 18, 296, 9506, 318752, 7221250, 39691136, 8488962, 512, 2
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OFFSET
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0,3
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COMMENTS
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The Stanley reference below describes a family of binomial posets whose elements are two colored graphs with vertices labeled on [n] and with edges labeled on [k-1]. T(n,k) is the number of elements in an n-interval of such a binomial poset. - Geoffrey Critzer, Aug 21 2023
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Volume 1, Second Edition, Example 3.18.3(e), page 323.
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LINKS
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FORMULA
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E.g.f. of column k: Sum_{j>=0} exp(k^j * x) * x^j/j!.
G.f. of column k: Sum_{j>=0} x^j/(1 - k^j * x)^(j+1).
For k>=1, E(x)^2 = Sum_{n>=0} T(n,k)*x^n/B_k(n) where B_k(n) = n!*k^binomial(n,2) and E(x) = Sum_{n>=0} x^n/b_k(n). - Geoffrey Critzer, Aug 21 2023
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, ...
2, 4, 6, 8, 10, 12, ...
2, 8, 26, 56, 98, 152, ...
2, 16, 162, 704, 2050, 4752, ...
2, 32, 1442, 15392, 84482, 318752, ...
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PROG
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(PARI) T(n, k) = sum(j=0, n, k^(j*(n-j))*binomial(n, j));
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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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