A226065 -id:A226065 - cp's OEIS frontend

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

Alford Arnold, Dec 04 2003

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
From Manfred Boergens, Jun 23 2024: (Start)
In the following two comments the restriction k<=n can be lifted, allowing all k>=1.
T(n,k) is the number of n X k binary matrices with row sums = 1.
T(n,k) is the number of coverings of [n] by tuples (A_1,...,A_k) in P([n])^k with disjoint A_j, with P(.) denoting the power set.
For nonempty A_j see A019538.
For tuples with "disjoint" dropped see A092477.
For tuples with nonempty A_j and with "disjoint" dropped see A218695. (End)

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

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).

Related to triangle of Eulerian numbers A008292.
Cf. A000312, A007778, A008788, A031971 (row sums), A062206, A083282.
Cf. A085524, A085526, A120485, A226065, A352981, A252982.
Cf. A019538, A092477, A218695.

a089072 = flip (^)
a089072_row n = map (a089072 n) [1..n]
a089072_tabl = map a089072_row [1..]  -- Reinhard Zumkeller, Mar 18 2013

[k^n: k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 01 2022

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

flatten([[k^n for k in range(1,n+1)] for n in range(1,12)]) # G. C. Greubel, Nov 01 2022

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)

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

Offset corrected by Reinhard Zumkeller, Mar 18 2013

A062815 a(n) = Sum_{i=1..n} i^(i+1).

Original entry on oeis.org

1, 9, 90, 1114, 16739, 296675, 6061476, 140279204, 3627063605, 103627063605, 3242055440326, 110235260819398, 4047611646518687, 159615707204330911, 6728024062917221536, 301875929242270047392, 14364960381309995038401

Offset: 1

Olivier Gérard, Jun 23 2001

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A226065, A226140.

Partial sums of A007778. - Michel Marcus, Mar 26 2019

Accumulate[#^(#+1)&/@Range[17]]  (* Harvey P. Dale, Mar 18 2011 *)

a(n) = sum(i=1, n, i^(i+1)); \\ Michel Marcus, Mar 26 2019

Definition simplified by Jon E. Schoenfield, Nov 29 2008

A226140 a(n) = Sum_{i=1..floor(n/2)} (n-i)^i.

Original entry on oeis.org

0, 1, 2, 7, 13, 48, 95, 424, 898, 4837, 10780, 68399, 158111, 1156224, 2745145, 22744380, 55098660, 510307001, 1255610350, 12859037607, 32030878113, 359498491968, 904385401323, 11040700820704, 28000658588542

Offset: 1

Wesley Ivan Hurt, May 27 2013

a(n) is the sum of the larger parts raised to the corresponding smaller parts of the partitions of n into exactly two parts.

			a(6) = 48; 6 has exactly 3 partitions into two parts: (5,1),(4,2),(3,3). Raising the larger parts to their corresponding smaller parts and adding the results, we get: 5^1 + 4^2 + 3^3 = 5 + 16 + 27 = 48.

Michael De Vlieger, Table of n, a(n) for n = 1..773
Index entries for sequences related to partitions

Cf. A226065.

A226140:=n->sum((n-i)^i, i=1..n/2): seq(A226140(n), n=1..40);

Array[Sum[(# - i)^i, {i, Floor[#/2]}] &, 25] (* Michael De Vlieger, Jan 23 2018 *)
Table[Total[#[[1]]^#[[2]]&/@IntegerPartitions[n,{2}]],{n,30}] (* Harvey P. Dale, Mar 25 2025 *)

a(n) = sum(k=1, n\2, (n-k)^k); \\ Michel Marcus, Dec 13 2015

a(n) = Sum_{i=1..floor(n/2)} (n-i)^i.

cp's OEIS Frontend

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

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A062815 a(n) = Sum_{i=1..n} i^(i+1).

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A226140 a(n) = Sum_{i=1..floor(n/2)} (n-i)^i.

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