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A330965
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Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).
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10
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1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 10, 20, 14, 1, 5, 14, 35, 70, 42, 1, 6, 18, 50, 126, 252, 132, 1, 7, 22, 65, 182, 462, 924, 429, 1, 8, 26, 80, 238, 672, 1716, 3432, 1430, 1, 9, 30, 95, 294, 882, 2508, 6435, 12870, 4862, 1, 10, 34, 110, 350, 1092, 3300, 9438, 24310, 48620, 16796
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OFFSET
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0,5
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
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LINKS
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FORMULA
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A(n,k) = (1 + k*n)*binomial(2*n,n)/(n+1).
A(n,k) = 2*(k*n+1)*(2*n-1)*A(n-1,k)/((n+1)*(k*n-k+1)) for n > 0.
G.f. of column k: (k - 1 - (2*k-4)*x - (k-1)*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)).
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EXAMPLE
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Array begins:
====================================================
n\k | 0 1 2 3 4 5 6 7
----+-----------------------------------------------
0 | 1 1 1 1 1 1 1 1 ...
1 | 1 2 3 4 5 6 7 8 ...
2 | 2 6 10 14 18 22 26 30 ...
3 | 5 20 35 50 65 80 95 110 ...
4 | 14 70 126 182 238 294 350 406 ...
5 | 42 252 462 672 882 1092 1302 1512 ...
6 | 132 924 1716 2508 3300 4092 4884 5676 ...
7 | 429 3432 6435 9438 12441 15444 18447 21450 ...
...
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PROG
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(PARI) T(n, k)={(1 + k*n)*binomial(2*n, n)/(n+1)}
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CROSSREFS
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Columns k=0..12 are A000108, A000984, A001700, A051924, A051944, A051945, A050476, A050477, A050478, A050479, A050489, A050490, A050491.
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KEYWORD
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AUTHOR
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STATUS
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approved
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