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A099020
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Euler-Seidel matrix T(k,n) with start sequence A001147, read by antidiagonals.
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5
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1, 1, 0, 2, 1, 1, 4, 2, 1, 0, 10, 6, 4, 3, 3, 26, 16, 10, 6, 3, 0, 76, 50, 34, 24, 18, 15, 15, 232, 156, 106, 72, 48, 30, 15, 0, 764, 532, 376, 270, 198, 150, 120, 105, 105, 2620, 1856, 1324, 948, 678, 480, 330, 210, 105, 0, 9496, 6876, 5020, 3696, 2748, 2070, 1590, 1260, 1050, 945, 945
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OFFSET
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0,4
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COMMENTS
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In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences, with the first column the inverse binomial transform of the start sequence.
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LINKS
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FORMULA
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Recurrence: T(0, 2n) = (2n-1)!!, T(0, 2n+1) = 0, T(k, n) = T(k-1, n) + T(k-1, n+1).
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EXAMPLE
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1, 0, 1, 0, 3, 0, 15, ...
1, 1, 1, 3, 3, 15, 15, ...
2, 2, 4, 6, 18, 30, 120, ...
4, 6, 10, 24, 48, 150, 330, ...
10, 16, 34, 72, 198, 480, 1590, ...
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MAPLE
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T:= proc(k, n) option remember; `if`(k=0, `if`(irem(n, 2)=0,
doublefactorial(n-1), 0), T(k-1, n) +T(k-1, n+1))
end:
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MATHEMATICA
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t[0, n_?EvenQ] := (n-1)!!; t[0, n_?OddQ] := 0; t[k_, n_] := t[k, n] = t[k-1, n] + t[k-1, n+1]; Table[t[k-n, n], {k, 0, 10}, {n, 0, k}] // Flatten (* Jean-François Alcover, Dec 10 2012 *)
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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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