A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.

1, 1, 4, 1, 8, 27, 1, 16, 81, 256, 1, 32, 243, 1024, 3125, 1, 64, 729, 4096, 15625, 46656, 1, 128, 2187, 16384, 78125, 279936, 823543, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489

Offset

1,3

Comments

T(n, k) = number of mappings from an n-element set into a k-element set. - Clark Kimberling, Nov 26 2004

Let S be the semigroup of (full) transformations on [n]. Let a be in S with rank(a) = k. Then T(n,k) = |a S|, the number of elements in the right principal ideal generated by a. - Geoffrey Critzer, Dec 30 2021

Links

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.1(ii).

Formula

Sum_{k=1..n} T(n, k) = A031971(n).

T(n, n) = A000312(n).

T(2*n, n) = A062206(n).

a(n) = (n + T*(1-T)/2)^T, where T = round(sqrt(2*n),0). - Gerald Hillier, Apr 12 2015

T(n,k) = A051129(n,k). - R. J. Mathar, Dec 10 2015

T(n,k) = Sum_{i=0..k} Stirling2(n,i)*binomial(k,i)*i!. - Geoffrey Critzer, Dec 30 2021

From G. C. Greubel, Nov 01 2022: (Start)

T(n, n-1) = A007778(n-1), n >= 2.

T(n, n-2) = A008788(n-2), n >= 3.

T(2*n+1, n) = A085526(n).

T(2*n-1, n) = A085524(n).

T(2*n-1, n-1) = A085526(n-1), n >= 2.

T(3*n, n) = A083282(n).

Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A120485(n).

Sum_{k=1..floor(n/2)} T(n-k, k) = A226065(n).

Sum_{k=1..floor(n/2)} T(n, k) = A352981(n).

Sum_{k=1..floor(n/3)} T(n, k) = A352982(n). (End)

Example

Triangle begins:
  1;
  1,  4;
  1,  8,  27;
  1, 16,  81,  256;
  1, 32, 243, 1024,  3125;
  1, 64, 729, 4096, 15625, 46656;
  ...

Mathematica

Column[Table[k^n, {n, 8}, {k, n}], Center] (* Alonso del Arte, Nov 14 2011 *)

Prog

(Haskell)
a089072 = flip (^)
a089072_row n = map (a089072 n) [1..n]
a089072_tabl = map a089072_row [1..]  -- Reinhard Zumkeller, Mar 18 2013
(Magma) [k^n: k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 01 2022
(SageMath) flatten([[k^n for k in range(1, n+1)] for n in range(1, 12)]) # G. C. Greubel, Nov 01 2022

Crossrefs

Related to triangle of Eulerian numbers A008292.

Cf. A000312, A007778, A008788, A031971 (row sums), A062206, A083282.

Cf. A085524, A085526, A120485, A226065, A352981, A252982.

Sequence in context: A077910 A366399 A100235 * A348595 A036177 A360131

Adjacent sequences: A089069 A089070 A089071 * A089073 A089074 A089075

Keyword

easy,nonn,tabl

Author

Alford Arnold, Dec 04 2003

Extensions

More terms and better definition from Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

Offset corrected by Reinhard Zumkeller, Mar 18 2013

Status

approved