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A086270
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Rectangular array T(k,n) of polygonal numbers, by antidiagonals.
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18
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1, 3, 1, 6, 4, 1, 10, 9, 5, 1, 15, 16, 12, 6, 1, 21, 25, 22, 15, 7, 1, 28, 36, 35, 28, 18, 8, 1, 36, 49, 51, 45, 34, 21, 9, 1, 45, 64, 70, 66, 55, 40, 24, 10, 1, 55, 81, 92, 91, 81, 65, 46, 27, 11, 1, 66, 100, 117, 120, 112, 96, 75, 52, 30, 12, 1, 78, 121, 145, 153, 148, 133, 111
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OFFSET
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1,2
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COMMENTS
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The antidiagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.
By rows, the sequence beginning (1, N, ...) is the binomial transform of (1, (N-1), (N-2), 0, 0, 0, ...); and is the second partial sum of (1, (N-2), (N-2), (N-2), ...). Example: The sequence (1, 4, 9, 16, 25, ...) is the binomial transform of (1, 3, 2, 0, 0, 0, ...) and the second partial sum of (1, 2, 2, 2, ...). - Gary W. Adamson, Aug 23 2015
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LINKS
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FORMULA
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T(n, k) = n*binomial(k, 2) + k = A057145(n+2,k).
2*T(n, k) = T(n+r, k) + T(n-r, k), where r = 0, 1, 2, 3, ..., n-1 (see table in Example field). - Bruno Berselli, Dec 19 2014
G.f.: x*y*(1 - x + x*y)/((1 - x)^2*(1 - y)^3).
G.f. of k-th column: k*(1 + k - 2*x)*x/(2*(1 - x)^2). (End)
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EXAMPLE
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First 6 rows:
=========================================
n\k| 1 2 3 4 5 6 7
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1 | 1 3 6 10 15 21 28 ... (A000217, triangular numbers)
2 | 1 4 9 16 25 36 49 ... (A000290, squares)
3 | 1 5 12 22 35 51 70 ... (A000326, pentagonal numbers)
4 | 1 6 15 28 45 66 91 ... (A000384, hexagonal numbers)
5 | 1 7 18 34 55 81 112 ... (A000566, heptagonal numbers)
6 | 1 8 21 40 65 96 133 ... (A000567, octagonal numbers)
...
The array formed by the complements: A183225.
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MATHEMATICA
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t[n_, k_] := n*Binomial[k, 2] + k; Table[ t[k, n - k + 1], {n, 12}, {k, n}] // Flatten
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PROG
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(Magma) T:=func<h, i | h*Binomial(i, 2)+i>; [T(k, n-k+1): k in [1..n], n in [1..12]]; // Bruno Berselli, Dec 19 2014
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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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