A247358 -id:A247358 - cp's OEIS frontend

A247363 Central terms of triangle A247358.

1, 3, 16, 64, 343, 4096, 59049, 1000000, 14348907, 129140163, 1475789056, 38443359375, 1099511627776, 34271896307633, 1156831381426176, 42052983462257059, 1152921504606846976, 18446744073709551616, 295147905179352825856, 12116574790945106558976

Offset: 1

Reinhard Zumkeller, Sep 14 2014

nonn

For all n there exist b(n) and e(n) such that: a(n)=b(n)^e(n) and b(n)+e(n)=2*n, see example.

			.   n |        a(n) | b(n)^e(n) | b(n)+e(n)
. ----+-------------+-----------+----------
.   1 |           1 |    1^1    |     2
.   2 |           3 |    3^1    |     4
.   3 |          16 |    2^4    |     6
.   4 |          64 |    2^6    |     8
.   5 |         343 |    7^3    |    10
.   6 |        4096 |    8^4    |    12
.   7 |       59049 |    9^5    |    14
.   8 |     1000000 |   10^6    |    16
.   9 |    14348907 |    3^15   |    18
.  10 |   129140163 |    3^17   |    20
.  11 |  1475789056 |   14^8    |    22
.  12 | 38443359375 |   15^9    |    24

Reinhard Zumkeller, Table of n, a(n) for n = 1..400

Cf. A247358, A000169.

Haskell
```
a247363 n = a247358 (2 * n - 1) n
```

row(n) = vecsort(vector(n, k, k^(n-k+1))); \\ A247358
a(n) = row(2*n-1)[n]; \\ Michel Marcus, Jan 24 2022

def A247363(n):
    return(sorted((b+1)**((2*n-1)-b) for b in range(2*n-1))[n-1])
# Chai Wah Wu, Sep 14 2014

a(n) = A247358(2*n-1,n).

A003101 a(n) = Sum_{k = 1..n} (n - k + 1)^k.

Original entry on oeis.org

0, 1, 3, 8, 22, 65, 209, 732, 2780, 11377, 49863, 232768, 1151914, 6018785, 33087205, 190780212, 1150653920, 7241710929, 47454745803, 323154696184, 2282779990494, 16700904488705, 126356632390297, 987303454928972, 7957133905608836, 66071772829247409

Offset: 0

N. J. A. Sloane, Henry W. Gould

For n > 0: a(n) = sum of row n of triangles A051129 and A247358. - Reinhard Zumkeller, Sep 14 2014
a(n-1) is the number of set partitions of [n] into two or more blocks such that all absolute differences between least elements of consecutive blocks are 1. a(3) = 8: 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, May 22 2017
Min_{n >= 1} a(n+1)/a(n) = 8/3. This is the answer to the 4th problem proposed during the first day of the final round of the 16th Austrian Mathematical Olympiad in 1985 (see link IMO Compendium). - Bernard Schott, Jan 07 2019
If n rings of different internal diameter can fit close together on a tapering column, a(n) is the number of different arrangements of at least one ring. For example, if the rings increasing in size are 1, 2 and 3, then a(3) = 8 corresponding to the possible arrangements from the point on the column of smallest diameter (1XX), (X2X), (XX3), (12X), (32X), (1X3), (X23) and (123), where X denotes a space on the column. - Ian Duff, Jun 23 2025

			For n = 3 we get a(3) = 3^1 + 2^2 + 1^3 = 8. For n = 4 we get a(4) = 4^1 + 3^2 + 2^3 + 1^4 = 22.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 0..598
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
Mathematics Stack Exchange, Asymptotics of 1^n+2^(n-1)+3^(n-2)+...+(n-1)^2+n^1, 2011.
Daniel Ropp, Problem 2 - 16th Austrian Mathematical Olympiad (Final round), Crux Mathematicorum, page 7, Vol. 14, Jun. 88.
The IMO compendium, Problem 2, 16th Austrian Mathematical Olympiad, 1985.
Index to sequences related to Olympiads.

Cf. A026898, A051129, A062810, A247358, A287215.

First differences are in A047970.

a003101 n = sum $ zipWith (^) [0 ..] [n + 1, n .. 1]
-- Reinhard Zumkeller, Sep 14 2014

[n eq 0 select 0 else (&+[(n-j+1)^j: j in [1..n]]): n in [0..50]]; // G. C. Greubel, Oct 26 2022

A003101 := n->add((n-k+1)^k, k=1..n);
a:= n-> add((n-j+1)^j, j=1..n): seq(a(n), n=0..30); # Zerinvary Lajos, Jun 07 2008

Table[Sum[(n-k+1)^k,{k,n}],{n,0,25}] (* Harvey P. Dale, Aug 14 2011 *)

a(n)=sum(k=1,n,(n-k+1)^k) \\ Charles R Greathouse IV, Oct 31 2011

def A003101(n): return sum( (n-k+1)^k for k in range(1,n+1))
[A003101(n) for n in range(50)] # G. C. Greubel, Oct 26 2022

a(n) = A026898(n) - 1.
G.f.: G(0)/x-1/(1-x)/x where G(k) = 1 + x*(2*k*x-1)/((2*k*x+x-1) - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
G.f.: Sum_{k>=1} x^k/(1 - (k + 1)*x). - Ilya Gutkovskiy, Oct 09 2018
a(n) = n^1 + (n-1)^2 + (n-2)^3 + ... + 3^(n-2) + 2^(n-1) + 1^n. - Bernard Schott, Jan 07 2019
log(a(n)) ~ (1 - 1/LambertW(exp(1)*n)) * n * log(1 + n/LambertW(exp(1)*n)). - Vaclav Kotesovec, Jun 15 2021
a(n) ~ sqrt(2*Pi/(n+1 + w(n))) * w(n)^(n+2 - w(n)), where w(n) = (n+1)/LambertW(exp(1)*(n+1)). - Vaclav Kotesovec, Jun 25 2021, after user "leonbloy", see Mathematics Stack Exchange link.

A051129 Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1).

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 8, 9, 4, 1, 16, 27, 16, 5, 1, 32, 81, 64, 25, 6, 1, 64, 243, 256, 125, 36, 7, 1, 128, 729, 1024, 625, 216, 49, 8, 1, 256, 2187, 4096, 3125, 1296, 343, 64, 9, 1, 512, 6561, 16384, 15625, 7776, 2401, 512, 81, 10, 1, 1024, 19683, 65536, 78125, 46656, 16807, 4096, 729, 100, 11

Offset: 1

N. J. A. Sloane

(n-th term) = (n-th term of A002260)^(n-th term of A004736). Both A002260 and A004736 are related to A002024. - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002

			  1   2       3       4       5       6       7
  1   4       9      16      25      36      49
  1   8      27      64     125     216     343
  1  16      81     256     625    1296    2401
  1  32     243    1024    3125    7776   16807
  1  64     729    4096   15625   46656  117649
  1 128    2187   16384   78125  279936  823543

T. D. Noe, Rows n = 1..50 of triangle, flattened

Cf. A051128 (transposed), A003992 (transposed), A004248.
Cf. A002024, A002260, A004736.
Cf. A002260, A003101 (antidiagonal sums), A000169 (central terms), A003320 (row maxima), A247358 (sorted rows).

a051129 n k = k ^ (n - k)
a051129_row n = a051129_tabl !! (n-1)
a051129_tabl = zipWith (zipWith (^)) a002260_tabl $ map reverse a002260_tabl
-- Reinhard Zumkeller, Sep 14 2014

T:= (n, k)-> k^n:
seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # Alois P. Heinz, Apr 18 2020

Table[ k^(n-k+1), {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 30 2012 *)

b(n) = floor(1/2 + sqrt(2 * n));
vector(100, n, (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1)) \\ Altug Alkan, Dec 09 2015

a(n) = (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1), where b(n) = [ 1/2 + sqrt(2 * n) ]. (b(n) is the n-th term of A002024.) - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002

More terms from James Sellers, Dec 11 1999

A003320 a(n) = max_{k=0..n} k^(n-k).

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 27, 81, 256, 1024, 4096, 16384, 78125, 390625, 1953125, 10077696, 60466176, 362797056, 2176782336, 13841287201, 96889010407, 678223072849, 4747561509943, 35184372088832, 281474976710656, 2251799813685248

Offset: 0

N. J. A. Sloane, R. K. Guy

For n > 0, a(n+1) = largest term of row n in triangles A051129 and A247358. - Reinhard Zumkeller, Sep 14 2014

			a(5) = max(5^0, 4^1, 3^2, 2^3, 1^4, 0^5) = max(1,4,9,8,1,0) = 9.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Tomescu, Introducere in Combinatorica. Editura Tehnica, Bucharest, 1972, p. 231.

Seiichi Manyama, Table of n, a(n) for n = 0..599 (terms 0..100 from T. D. Noe).
D. Easdown, Minimal faithful permutation and transformation representations of groups and semigroups, Contemporary Math. (1992), Vol. 131 (Part 3), 75-84.
R. Gray and J. D. Mitchell, Largest subsemigroups of the full transformation monoid, Discrete Math., 308 (2008), 4801-4810.
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
R. K. Guy, Letter to N. J. A. Sloane, Mar 1974
I. Tomescu, Excerpts from "Introducese in Combinatorica" (1972), pp. 230-1, 44-5, 128-9. (Annotated scanned copy)

Cf. A003992, A031435, A056155.

a003320 n = maximum $ zipWith (^) [0 .. n] [n, n-1 ..]
-- Reinhard Zumkeller, Jun 24 2013

Join[{1},Max[#]&/@Table[k^(n-k),{n,25},{k,n}]] (* Harvey P. Dale, Jun 20 2011 *)

a(n) = vecmax(vector(n+1, k, (k-1)^(n-k+1))); \\ Michel Marcus, Jun 13 2017

a(n) = A056155(n-1)^(n - A056155(n-1)), for n >= 2. - Ridouane Oudra, Dec 09 2020

Easdown reference from Michail Kats (KatsMM(AT)info.sgu.ru)

More terms from James Sellers, Aug 21 2000

cp's OEIS Frontend

A247363 Central terms of triangle A247358.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Python

Formula

A003101 a(n) = Sum_{k = 1..n} (n - k + 1)^k.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

SageMath

Formula

A051129 Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A003320 a(n) = max_{k=0..n} k^(n-k).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions