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

0,5

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

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].

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

Example

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

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 *)

Prog

(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

Crossrefs

Row sums give A026898.

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

Cf. A095884, A051128.

Cf. A009998 (mirrored).

Sequence in context: A080853 A071922 A138028 * A322268 A144823 A098446

Adjacent sequences: A009996 A009997 A009998 * A010000 A010001 A010002

Keyword

tabl,nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved