A075363 -id:A075363 - cp's OEIS frontend

A075364 a(n) = floor( geometric mean of n-th row of A075363).

1, 2, 9, 32, 125, 529, 2401, 11585, 59049, 316227, 1771561, 10343751, 62748517, 394421215, 2562890625, 17179869184, 118587876497, 841567195983, 6131066257801, 45794672179195, 350277500542221, 2740695769692949

Offset: 1

Amarnath Murthy, Sep 20 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..500 (terms 1..100 from Vincenzo Librandi)

Cf. A075363, A147771, A190343.

[Floor(n^((n+1)/2)): n in [1..30]]; // Vincenzo Librandi, May 07 2011

Table[Floor[n^((n + 1)/2)], {n,1,50}] (* G. C. Greubel, Dec 30 2017 *)

for(n=1, 30, print1(floor(n^((n+1)/2)), ", ")) \\ G. C. Greubel, Dec 30 2017

from math import isqrt
def A075364(n): return isqrt(n**(n+1)) # Chai Wah Wu, Jun 08 2025

a(n) = floor(n^((n+1)/2)).

More terms from Sascha Kurz, Jan 12 2003

A068424 Triangle of falling factorials, read by rows: T(n, k) = n(n-1)...*(n-k+1), n > 0, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 4, 12, 24, 24, 5, 20, 60, 120, 120, 6, 30, 120, 360, 720, 720, 7, 42, 210, 840, 2520, 5040, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 362880, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800

Offset: 1

David Wasserman, Mar 13 2003

Triangle in which the n-th row begins with n and the k-th term is obtained by multiplying the (k-1)-th term by (n-k+1) until n-k+1 = 1. - Amarnath Murthy, Nov 11 2002
Has many diagonals in common with A105725. - Zerinvary Lajos, Apr 14 2006
Also: the square array of rising factorials A(n,k) = n*(n+1)*(n+2)*...*(n+k-1) read by antidiagonals downwards. There are no perfect squares in T(n,k) for k >= 2 [see Rigge]. T(n,k) is divisible by a prime exceeding k, if n >= 2*k [see Saradha and Shorey]. - R. J. Mathar, May 02 2007
T(n,k) is the number of injective functions f from {1,...,k} into {1,...,n}, since there are n choices for f(1), then (n-1) choices for f(2), ... and (n-k+1) choices for f(k). E.g., T(3,2)=6 because there are exactly 6 injective functions f:{1,2}->{1,2,3}, namely, f1={(1,1),(2,2)}, f2={(1,1),(2,3)}, f3={(1,2),(2,1)}, f4={(1,2),(2,3)}, f5={(1,3),(2,1)} and f6={(1,3),(2,2)}. - Dennis P. Walsh, Oct 18 2007
Permuted words defined by the connectivity of regular simplices are related to T by T = A135278 * (1!, 2!, 3!, 4!, ...). E.g., for T(4,k) with k-1 = simplex number, label the vertices of a tetrahedron with a, b, c, d, then the 0-simplex, the points, a,b,c,d gives 4 * 1 = 4 words; the 1-simplex, the edges: (ab or ba), (ac or ca), (ad or da), (bc or cb), (bd or db), (cd or dc) gives 6 * 2 = 12 words; the 2-simplex, the faces: (abc or ...), (acd or ...), (adb or ...), (bcd or ...) gives 4 * 6 = 24 words; the 3-simplex, (abcd or ....) gives 1 * 24 = 24 words. - Tom Copeland, Dec 08 2007
Reversal of the triangle by rows = (n+1) * n-th row of triangle A094587. - Gary W. Adamson, May 03 2009
From Geoffrey Critzer, May 06 2009: (Start)
The rectangular array R(n,k), read by diagonals is the number of ways n people can queue up in k (possibly empty) distinct queues. R(n,k) = (n+k-1)!/(k-1)!; R(n,k) = (n+k-1)*R(n-1,k).
Northwest corner:
    1,   2,    3,    4,     5, ...;
    2,   6,   12,   20,    30, ...;
    6,  24,   60,  120,   210, ...;
   24, 120,  360,  840,  1680, ...;
  120, 720, 2520, 6720, 15120, ...;
  ...
Example: R(2,2)=6 because there are six ways that two people can get in line at a fast food restaurant that has two order windows open. Let 1 and 2 represent the two people and a | will separate the lines. 12|; 21|; |12; |21; 1|2; 2|1. (End)
Cf. [Hardy and Wright], Theorem 34.
The e.g.f. of the Norlund generalized Bernoulli (Appell) polynomials of order m, NB(n,x;m), is given by exponentiation of the e.g.f. of the Bernoulli numbers, i.e., multiple binomial self-convolutions of the Bernoulli numbers, through the e.g.f. exp[NB(.,x;m)t] = [t/(e^t-1)]^(m+1) * e^(xt). Norlund gave the relation to the factorials (x-1)!/(x-1-k)! = (x-1) ... (x-k) = NB(k,x;k), so T(n,k) = NB(k,n+1;k). - Tom Copeland, Oct 01 2015
T(n,k) is the number of sequences without repetition (partial permutations) of k elements taken from an n-set. For sequences with repetition (k-tuples) cf. A075363. - Manfred Boergens, Jun 18 2023

			Triangle begins:
  1;
  2,  2;
  3,  6,   6;
  4, 12,  24,  24;
  5, 20,  60, 120, 120;
  6, 30, 120, 360, 720, 720;
Square begins:
    1,   2,    3,    4,     5, ...
    2,   6,   12,   20,    30, ...
    6,  24,   60,  120,   210, ...
   24, 120,  360,  840,  1680, ...
  120, 720, 2520, 6720, 15120, ...

G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Fifth edition, 1979, p. 64.
O. Rigge, 9th Congr. Math. Scan., Helsingfors, 1938, Mercator, 1939, pp. 155-160.

G. C. Greubel, Table of n, a(n) for the first 100 rows, 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 (iii), p. 131.
N. Saradha and T. N. Shorey, Almost Squares and Factorisations in Consecutive Integers, Compositio Mathematica 138 (1) (2003) 113-124.
Eric Weisstein's World of Mathematics, Falling Factorial.

Cf. A007318, A000142, A075363.
Same as A008279 for k>0.
Cf. A094587. - Gary W. Adamson, May 03 2009
Appears in A167546. - Johannes W. Meijer, Nov 12 2009

Flatten[Table[n!/(n-k)!, {n, 10}, {k, n}]] (* or, from version 7: *)
Flatten[Table[FactorialPower[n, k], {n, 10}, {k, n}]]  (* Jean-François Alcover, Jun 17 2011, updated Sep 29 2016 *)

PARI
```
T(n,k)=if(k<1 || k>n,0,n!/(n-k)!)
```

As a triangle: T(n,k) = k!*binomial(n,k) = n!/(n-k)!, 1 <= k <= n. - Michael Somos, Apr 05 2003
E.g.f.: exp(x)*x*y/(1-x*y). - Michael Somos, Apr 05 2003
As a square: A(n,k) = (n+k-1)!/(k-1)!, 1 <= k <= n. - Ron L.J. van den Burg, Nov 28 2021

A090414 Triangular array read by rows, 1<=k<=n: T(n,k) = smallest prime that in binary representation is not contained in n^k.

Original entry on oeis.org

2, 3, 3, 2, 3, 7, 3, 3, 3, 3, 3, 5, 11, 5, 7, 5, 3, 7, 3, 5, 7, 2, 5, 13, 7, 11, 13, 17, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 7, 7, 11, 19, 19, 17, 13, 3, 5, 11, 5, 7, 11, 7, 17, 17, 13, 7, 5, 7, 5, 23, 7, 13, 7, 7, 17, 17, 5, 3, 7, 3, 5, 7, 7, 7, 5, 11, 17, 19

Offset: 1

Reinhard Zumkeller, Nov 30 2003

			Triangle begins:
2;
3, 3;
2, 3,  7;
3, 3,  3, 3;
3, 5, 11, 5,  7;
5, 3,  7, 3,  5,  7;
2, 5, 13, 7, 11, 13, 17;
..

Jinyuan Wang, Rows n = 1..100 of triangle, flattened

Cf. A004676, A007088, A075363.

T(n,k) = {my(d=binary(n^k), v); forprime(p=2, oo, v=binary(p); if(#v>#d || sum(i=1, #d-#v+1, v==vector(#v, j, d[i+j-1]))==0, return(p))); } \\ Jinyuan Wang, Mar 01 2020

More terms from Jinyuan Wang, Mar 01 2020

A099615 Triangle read by rows, 2<=k<=n: T(n,k) = denominator of (1+1/n)^k-(1+k/n) and of (1-1/n)^k-(1-k/n).

Original entry on oeis.org

4, 9, 27, 16, 64, 256, 25, 125, 625, 3125, 36, 216, 1296, 7776, 46656, 49, 343, 2401, 16807, 117649, 823543, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 100, 1000, 10000, 100000

Offset: 1

Reinhard Zumkeller, Oct 25 2004

Corresponding numerators: A099613, A099614.

Equals A075363 except for the leading column (k = 1) in that triangle. - Georg Fischer, Jul 26 2023

Cf. A075363, A099613, A099614.

T(n, k) = n^k.

A326120 a(n) is the concatenation of n^1, n^2, ..., n^n.

Original entry on oeis.org

1, 24, 3927, 41664256, 5251256253125, 6362161296777646656, 749343240116807117649823543, 864512409632768262144209715216777216, 981729656159049531441478296943046721387420489, 10100100010000100000100000010000000100000000100000000010000000000

Offset: 1

Trevor Mulindi, Sep 10 2019

a(100) has 10200 digits.

a(n) is the concatenation of the n terms of the n-th row of A075363 triangle. - Michel Marcus, Sep 15 2019

Cf. A007908, A019521, A019522, A053062, A075363.

a:= n-> parse(cat(n^j$j=1..n)):
seq(a(n), n=1..14);  # Alois P. Heinz, Jan 24 2021

a[n_] := FromDigits@ Flatten@ IntegerDigits[n^Range[n]]; Array[a, 10] (* Giovanni Resta, Sep 12 2019 *)

def a(n): return int("".join(str(n**i) for i in range(1, n+1)))
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Jan 24 2021

More terms from Giovanni Resta, Sep 12 2019

A338280 Triangle T read by rows: T(n, k) = k*n^(n-k-1) with 0 < k < n.

Original entry on oeis.org

1, 3, 2, 16, 8, 3, 125, 50, 15, 4, 1296, 432, 108, 24, 5, 16807, 4802, 1029, 196, 35, 6, 262144, 65536, 12288, 2048, 320, 48, 7, 4782969, 1062882, 177147, 26244, 3645, 486, 63, 8, 100000000, 20000000, 3000000, 400000, 50000, 6000, 700, 80, 9, 2357947691, 428717762, 58461513, 7086244, 805255, 87846, 9317, 968, 99, 10

Offset: 2

Stefano Spezia, Oct 20 2020

T(n, k) is the number of forests of n - k edges that connect every other labeled vertex to one of the k roots (see Section 3 in Wästlund).

Alfred Rényi, Some remarks on the theory of trees. MTA Mat. Kut. Inst. Kozl. (Publ. math. Inst. Hungar. Acad. Sci) 4 (1959), 73-85.

Arthur Cayley, A theorem on trees, Quart. J. Pure Appl. Math. 23: 376-378 (1889). Also in The collected mathematical papers of Arthur Cayley vol 13.
John Riordan, Forests of labeled trees, Journal of Combinatorial Theory 5 (1968), 93-103.
Lajos Takács, On Cayley’s Formula for Counting Forests, Journal of Combinatorial Theory Series A 53, 321-323 (1990). See Equation 1.
Johan Wästlund, Padlock Solitaire: A martingale trick for combinatorial enumeration, arXiv:2008.13017 [math.CO], 2020. See Section 3.

Cf. A000027 (diagonal), A000169, A000272 (1st column), A000312, A007334 (2nd column), A023811 (row sums), A034941, A072590, A075363, A210725.

Table[k*n^(n-k-1),{n,2,11},{k,1,n-1}]//Flatten

cp's OEIS Frontend

A075364 a(n) = floor( geometric mean of n-th row of A075363).

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A068424 Triangle of falling factorials, read by rows: T(n, k) = n*(n-1)*...*(n-k+1), n > 0, 1 <= k <= n.

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A090414 Triangular array read by rows, 1<=k<=n: T(n,k) = smallest prime that in binary representation is not contained in n^k.

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A099615 Triangle read by rows, 2<=k<=n: T(n,k) = denominator of (1+1/n)^k-(1+k/n) and of (1-1/n)^k-(1-k/n).

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A326120 a(n) is the concatenation of n^1, n^2, ..., n^n.

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Mathematica

Python

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A338280 Triangle T read by rows: T(n, k) = k*n^(n-k-1) with 0 < k < n.

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A068424 Triangle of falling factorials, read by rows: T(n, k) = n(n-1)...*(n-k+1), n > 0, 1 <= k <= n.