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 A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n. 21
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified December 1 13:31 EST 2023. Contains 367475 sequences. (Running on oeis4.)