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A009999
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Triangle in which j-th entry in i-th row is (i+1-j)^j, 0<=j<=i.
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9
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1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 8, 1, 1, 5, 16, 27, 16, 1, 1, 6, 25, 64, 81, 32, 1, 1, 7, 36, 125, 256, 243, 64, 1, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1
(list;
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graph;
refs;
listen;
history;
text;
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OFFSET
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0,5
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COMMENTS
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T(n,k) is the number of ways of placing 1..k in n boxes such that each box contains at most one number, and numbers in adjacent boxes are in increasing order. This can be proved by observing that there are n-(k-1) ways of extending each of T(n-1,k-1). - Jimin Park, Apr 16 2023
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 24.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-1-i,k-i). - Jimin Park, Apr 16 2023
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EXAMPLE
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Triangle begins
1
1 1
1 2 1
1 3 4 1
1 4 9 8 1
1 5 16 27 16 1
1 6 25 64 81 32 1
1 7 36 125 256 243 64 1
1 8 49 216 625 1024 729 128 1
1 9 64 343 1296 3125 4096 2187 256 1
1 10 81 512 2401 7776 15625 16384 6561 512 1
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MAPLE
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MATHEMATICA
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PROG
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(Haskell)
a009999 n k = (n + 1 - k) ^ k
a009999_row n = a009999_tabl !! n
a009999_tabl = [1] : map snd (iterate f ([1, 1], [1, 1])) where
f (us@(u:_), vs) = (us', 1 : zipWith (*) us' vs)
where us' = (u + 1) : us
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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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