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%I #45 May 12 2024 17:34:35
%S 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,
%T 36,125,256,243,64,1,1,8,49,216,625,1024,729,128,1,1,9,64,343,1296,
%U 3125,4096,2187,256,1,1,10,81,512,2401,7776,15625,16384,6561,512,1
%N Triangle in which j-th entry in i-th row is (i+1-j)^j, 0<=j<=i.
%C 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
%C The n-th diagonal consists of n^k. This can also be generated as the Akiyama-Tanigawa algorithm applied to the sequence binomial(n+k,k). - _Shel Kaphan_, May 03 2024
%D 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.
%H Reinhard Zumkeller, <a href="/A009999/b009999.txt">Rows n = 0..125 of triangle, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F T(n,0) = 1; T(n,k) = (n-k+1)*T(n-1,k-1) for k=1..n. - _Reinhard Zumkeller_, Feb 02 2014
%F T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-1-i,k-i). - _Jimin Park_, Apr 16 2023
%e Triangle begins
%e 1
%e 1 1
%e 1 2 1
%e 1 3 4 1
%e 1 4 9 8 1
%e 1 5 16 27 16 1
%e 1 6 25 64 81 32 1
%e 1 7 36 125 256 243 64 1
%e 1 8 49 216 625 1024 729 128 1
%e 1 9 64 343 1296 3125 4096 2187 256 1
%e 1 10 81 512 2401 7776 15625 16384 6561 512 1
%p A009999 := proc(i,j) (i+1-j)^j ; end proc: # _R. J. Mathar_, Jan 16 2011
%t Table[(i+1-j)^j, {i, 0, 10}, {j, 0, i}] // Flatten (* _Jean-François Alcover_, Jan 14 2014 *)
%o (Haskell)
%o a009999 n k = (n + 1 - k) ^ k
%o a009999_row n = a009999_tabl !! n
%o a009999_tabl = [1] : map snd (iterate f ([1,1], [1,1])) where
%o f (us@(u:_), vs) = (us', 1 : zipWith (*) us' vs)
%o where us' = (u + 1) : us
%o -- _Reinhard Zumkeller_, Feb 02 2014
%Y Row sums give A026898.
%Y T(2n,n) gives A000169(n+1).
%Y Cf. A095884, A051128.
%Y Cf. A009998 (mirrored).
%K tabl,nonn,easy
%O 0,5
%A _N. J. A. Sloane_
%E T(10,8) corrected by _Reinhard Zumkeller_, Feb 02 2014
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