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A214398
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Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.
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6
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1, 1, 1, 1, 4, 1, 1, 10, 9, 1, 1, 20, 45, 16, 1, 1, 35, 165, 136, 25, 1, 1, 56, 495, 816, 325, 36, 1, 1, 84, 1287, 3876, 2925, 666, 49, 1, 1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1, 1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1, 1, 220, 12870, 170544
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OFFSET
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1,5
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COMMENTS
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This is also the array A(n,k) read upwards antidiagonals, where the entry in row n and column k counts the vertex-labeled digraphs with n arcs and k vertices, allowing multi-edges and multi-loops (labeled analog to A138107). The binomial formula counts the weak compositions of distributing n arcs over the k^2 positions in the adjacency matrix. - R. J. Mathar, Aug 03 2017
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LINKS
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FORMULA
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T(n,k) = binomial(k^2+n-k-1, n-k).
T(n,n-3) = (n^2-6*n+11)*(n^2-6*n+10)*(n-3)^2 /6. - R. J. Mathar, Aug 03 2017
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 4, 1;
1, 10, 9, 1;
1, 20, 45, 16, 1;
1, 35, 165, 136, 25, 1;
1, 56, 495, 816, 325, 36, 1;
1, 84, 1287, 3876, 2925, 666, 49, 1;
1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1;
1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1;
1, 220, 12870, 170544, 593775, 658008, 270725, 45760, 3321, 100, 1; ...
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MAPLE
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binomial(k^2+n-k-1, n-k) ;
end proc:
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MATHEMATICA
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nmax = 11;
T[n_, k_] := SeriesCoefficient[1/(1-x)^(k^2), {x, 0, n-k}];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten
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PROG
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(PARI) T(n, k)=binomial(k^2+n-k-1, n-k)
for(n=1, 11, for(k=1, n, print1(T(n, k), ", ")); print(""))
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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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