A009999 Triangle in which j-th entry in i-th row is (i+1-j)^j, 0<=j<=i.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• tabl
• nonn
• easy

Comments

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

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

Examples

			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
		

References

• 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.

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• 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].

Crossrefs

Row sums give A026898.

T(2n,n) gives A000169(n+1).

Cf. A095884, A051128.

Cf. A009998 (mirrored).

Programs

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
-- Reinhard Zumkeller, Feb 02 2014

Maple

A009999 := proc(i,j) (i+1-j)^j ; end proc: # R. J. Mathar, Jan 16 2011

Mathematica

Table[(i+1-j)^j, {i, 0, 10}, {j, 0, i}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

Formula

T(n,0) = 1; T(n,k) = (n-k+1)*T(n-1,k-1) for k=1..n. - Reinhard Zumkeller, Feb 02 2014

T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-1-i,k-i). - Jimin Park, Apr 16 2023

Extensions

T(10,8) corrected by Reinhard Zumkeller, Feb 02 2014