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.
KEYWORD
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