A076113 -id:A076113 - cp's OEIS frontend

A002489 a(n) = n^(n^2), or (n^n)^n.

1, 1, 16, 19683, 4294967296, 298023223876953125, 10314424798490535546171949056, 256923577521058878088611477224235621321607, 6277101735386680763835789423207666416102355444464034512896, 196627050475552913618075908526912116283103450944214766927315415537966391196809

Offset: 0

N. J. A. Sloane

The number of closed binary operations on a set of order n. Labeled groupoids.
The values of "googol" in base N: "10^100" in base 2 is 2^4=16; "10^100" in base 3 is 3^9=19683, etc. This is N^^3 by the "lower-valued" (left-associative) definition of the hyper4 or tetration operator (see Munafo webpage). - Robert Munafo, Jan 25 2010
n^(n^k) = (((n^n)^n)^...)^n, with k+1 n's, k >= 0. - Daniel Forgues, May 18 2013

			a(3) = 19683 because (3^3)^3 = 3^(3^2) = 19683.

John S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 6.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael Lee, Table of n, a(n) for n = 0..26 (first 16 terms from Vincenzo Librandi)
Robert Munafo, Hyper4 Iterated Exponential Function [From _Robert Munafo_, Jan 25 2010]
Eric Postpischil, Posting to sci.math newsgroup, May 21 1990.
P. Rossier, Grands nombres, Elemente der Mathematik, Vol. 3 (1948), p. 20; alternative link.
Index entries for sequences related to groupoids

a(n) = A079172(n) + A023814(n) = A079176(n) + A079179(n);
a(n) = A079182(n) + A023813(n) = A079186(n) + A079189(n);
a(n) = A079192(n) + A079195(n) + A079198(n) + A023815(n).
Cf. A002488, A001329, A002488, A023813, A076113, A090588, A000312, A258102.

[n^(n^2): n in [0..10]]; // Vincenzo Librandi, May 13 2011

Join[{1},Table[n^n^2,{n,10}]] (* Harvey P. Dale, Sep 06 2011 *)

a(n)=n^(n^2) \\ Charles R Greathouse IV, Nov 20 2012

a(n) = [x^(n^2)] 1/(1 - n*x). - Ilya Gutkovskiy, Oct 10 2017

Sum_{n>=1} 1/a(n) = A258102. - Amiram Eldar, Nov 11 2020

A355400 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}.

Original entry on oeis.org

1, 1, 3, 30, 1001, 111384, 41314284, 51067020290, 210309203300625, 2885318087540733000, 131857099297936066411200, 20070377346929658409924542720, 10174783866874800701945612292557712, 17178820188393063395267380511228827387600, 96592800670609299321035523895170598736583965100

Offset: 0

Alois P. Heinz, Jun 30 2022

nonn

Determinant of the n X n Hankel matrix whose i-th antidiagonal is filled with the n+i-th Catalan number for i = 0..2*n-2.
              [ 5,  14,  42]
  a(3) = det( [14,  42, 132] ) = 30.
              [42, 132, 429]

			a(0) = 1:  ( ).
a(1) = 1:  (/\).
a(2) = 3:                        /\      /\    /\
           (/\/\, /\/\), (/\/\, /  \), (/  \, /  \).
G.f. = 1 + x + 3*x^2 + 30*x^3 + 1001*x^4 + 111384*x^5 + 41314284*x^6 + ... - _Michael Somos_, Jun 27 2023

Alois P. Heinz, Table of n, a(n) for n = 0..62
Wikipedia, Counting lattice paths
Wikipedia, Hankel matrix

A diagonal of A078920 or of A123352 or of A368025.

Cf. A000108, A006125, A074962, A076113, A110131, A112332, A131577, A358597, A368298, A378113.

a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n-1), i=1..n-1):
seq(a(n), n=0..14);

Join[{1}, Table[Sqrt[2*BarnesG[4*n]] * BarnesG[n] * Gamma[2*n]^(3/2) / BarnesG[3*n + 1], {n, 1, 12}]] (* Vaclav Kotesovec, Aug 26 2023 *)

a(n) = prod(i=1, n-1, prod(j=i, n-1, (i+j+2*n)/(i+j))); \\ Michel Marcus, Jul 05 2022

a(n) = Product_{i=1..n-1, j=i..n-1} (i+j+2*n)/(i+j).
a(n) mod 2 = 1 <=> n in { A131577 }.
a(n) ~ exp(1/24) * 2^(1/6 - n + 8*n^2) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 - 1/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 26 2023

A090588 Number of labeled idempotent groupoids.

Original entry on oeis.org

1, 1, 4, 729, 16777216, 95367431640625, 221073919720733357899776, 311973482284542371301330321821976049, 374144419156711147060143317175368453031918731001856, 507528786056415600719754159741696356908742250191663887263627442114881

Offset: 0

Christian G. Bower, Dec 03 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..26
Index entries for sequences related to groupoids

Cf. A002489, A030247, A076113.

[n^(n^2 - n): n in [0..10]]; // Vincenzo Librandi, Aug 08 2015

a:=n->mul(mul(sum(1, j=1..n), k=1..n), m=1..n-1): seq(a(n), n=0..8); # Zerinvary Lajos, Dec 31 2008

Join[{1},Table[n^(n^2-n),{n,10}]] (* Harvey P. Dale, Sep 16 2013 *)

a(n) = n^(n^2-n); \\ Joerg Arndt, Nov 04 2013

def a(n)
ids =* (0..n-1)
return (ids.product(ids)).reduce(1){ |accum,x|  (x[0] == x[1]) ? accum :  accum*ids.length}
end
# Chad Brewbaker, Nov 03 2013

a(n) = n^(n^2 - n).

One additional term from Harvey P. Dale, Sep 16 2013

A030257 Number of nonisomorphic commutative idempotent groupoids.

Original entry on oeis.org

1, 1, 1, 7, 192, 82355, 653502972, 110826042515867, 479732982053513924168, 62082231641825701423422054735, 275573192431752191557427399293883120600, 47363301285150007842253190185182901101879369430257

Offset: 0

Christian G. Bower, Feb 15 1998, May 15 1998 and Dec 03 2003

nonn

Index entries for sequences related to groupoids

Cf. A001329, A038017, A038021, A076113.

a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = Product_{i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (Sum_{d|i} (d*s_d))^((i*s_i^2-s_i)/2) or {i=j, even} (Sum_{d|i} (d*s_d))^((i*s_i^2-2*s_i)/2) * (Sum_{d|i/2} (d*s_d))^s_i or {i != j} (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j). - Corrected by Sean A. Irvine, Mar 27 2020

a(n) is asymptotic to (n^binomial(n-1, 2))/n! = A076113(n)/A000142(n).

A038017 Number of n-element commutative groupoids with an identity ("pointed" groupoids).

Original entry on oeis.org

1, 2, 15, 720, 409600, 3920030472, 775775333825891, 3837862827737186253664, 558740081065710564284870598075, 2755731923933734753149997221152548428020, 520996314135332606285488148844494695722050333912483

Offset: 1

Christian G. Bower, May 15 1998; revised Dec 05 2003

nonn

Also number of commutative partial groupoids with n-1 elements or commutative groupoids with an absorbant (zero) element with n elements.

Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Eric Weisstein's World of Mathematics, Groupoid.
Index entries for sequences related to groupoids

Cf. A001329, A030257.

a(n+1) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (1 + sum {d|i} (d*s_d))^((i*s_i^2+s_i)/2) or {i=j, even} (1 + sum {d|i} (d*s_d))^(i*s_i^2/2) * (1 + sum {d|i/2} (d*s_d))^s_i or {i != j} (1 + sum {d|lcm(i, j)} (d*s_d))^(2*gcd(i, j)*s_i*s_j)

a(n) asymptotic to (n^binomial(n, 2)+1)/n! = A090599(n)/A000142(n) = A076113(n)/A000142(n-1)

A076112 Triangle (read by rows) in which the n-th row contains first n terms of n geometric progression with first term 1 and common ratio (n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 9, 1, 4, 16, 64, 1, 5, 25, 125, 625, 1, 6, 36, 216, 1296, 7776, 1, 7, 49, 343, 2401, 16807, 117649, 1, 8, 64, 512, 4096, 32768, 262144, 2097152, 1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 1, 10, 100, 1000, 10000, 100000, 1000000

Offset: 1

Amarnath Murthy, Oct 09 2002

Table T(n,k) = (n+k-1)^(n-1) n,k > 0 read by antidiagonals. - Boris Putievskiy, Dec 17 2012

			Triangle begins:
  1;
  1,2;
  1,3,9;
  1,4,16,64;
  1,5,25,125,625;
  1,6,36,216,1296,7776;
  ...

Harvey P. Dale, Table of n, a(n) for n = 1..5000
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A000169, A076113.

Table[n^Range[0,n-1],{n,10}]//Flatten (* Harvey P. Dale, Jan 27 2020 *)

from math import isqrt, comb
def A076112(n): return (isqrt(n<<3)+1>>1)**(n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2)-1) # Chai Wah Wu, Jun 09 2025

As a linear array, the sequence is a(n) = A002024(n)^A002260(n) or a(n) = (t+1)^(n-t*(t+1)/2-1), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

A355561 Number of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= n^(i-1).

Original entry on oeis.org

1, 1, 2, 24, 3236, 7173370, 330736663032, 382149784071841422, 12983632019302863224103688, 14912674110246473369128526689667934, 654972005961623890774153743504185499487372010, 1228018869478731662593970252736815943512232438560622483276

Offset: 0

Alois P. Heinz, Jul 06 2022

nonn

			a(0) = 1: ( ).
a(1) = 1: (1).
a(2) = 2: (1,1), (1,2).
a(3) = 24: (1,1,1), (1,1,2), (1,1,3), (1,1,4), (1,1,5), (1,1,6), (1,1,7), (1,1,8), (1,1,9), (1,2,2), (1,2,3), (1,2,4), (1,2,5), (1,2,6), (1,2,7), (1,2,8), (1,2,9), (1,3,3), (1,3,4), (1,3,5), (1,3,6), (1,3,7), (1,3,8), (1,3,9).

Alois P. Heinz, Table of n, a(n) for n = 0..36

Main diagonal of A355576.

Cf. A076113, A090588, A107354, A355519.

b:= proc(n, k, i) option remember; `if`(n=0, 1,
      add(b(n-1, k, j), j=1..min(i, k^(n-1))))
    end:
a:= n-> b(n$2, infinity):
seq(a(n), n=0..6);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, -add(
      b(j, k)*(-1)^(n-j)*binomial(k^j, n-j), j=0..n-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..12);

A090599 Number of n-element labeled commutative groupoids with an identity.

Original entry on oeis.org

1, 4, 81, 16384, 48828125, 2821109907456, 3909821048582988049, 154742504910672534362390528, 202755595904452569706561330872953769, 10000000000000000000000000000000000000000000000

Offset: 1

Christian G. Bower, Dec 05 2003

nonn

Also labeled commutative groupoids with an absorbant (zero) element.

Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Eric Weisstein's World of Mathematics, Groupoid.
Index entries for sequences related to groupoids

a(n) = A076113(n)*n. Cf. A038017.

a(n) = n^(1+binomial(n, 2))

A120929 Partial sums of n^(n^2), A002489.

Original entry on oeis.org

1, 2, 18, 19701, 4294986997, 298023228171940122, 10314424798788558774343889178, 256923577521069192513410265783009965210785, 6277101735386681020759366944276858929512621227473999723681

Offset: 0

Jonathan Vos Post, Aug 18 2006

After 2, can this ever be prime? This is to A001923 Sum k^k, k=1..n, as k^k^k is to k^k.

			a(0) = 1 because A002489(0) is given formally as 0^0^0 = 1.
a(1) = 2 because 1 + (1^1)^1 = 1 + 1 = 2.
a(2) = 18 because 2 + (2^2)^2 = 2 + 16 = 18.
a(3) = 19701 because 18 + (3^3)^3 = 18 + 19683 = 19701.
a(4) = 4294986997 = 19701 + (4^4)^4 = 19701 + 4294967296.

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A001923, A002489, A002488, A001329, A002488, A023813, A076113, A090588.

Accumulate[Join[{1},Table[n^(n^2),{n,9}]]] (* Harvey P. Dale, Apr 10 2014 *)

a(n) = Sum_{i=0..n} i^(i^2). a(n) = Sum_{i=0..n} (i^i)^i. In this sequence, we formally define 0^0 = 1.

More terms from Harvey P. Dale, Apr 10 2014

A385820 Number of equivalence classes of finitely-supported integer functions on Z^2 modulo moves that add + or -1 to every cell whose coordinates form an arithmetic progression of length n.

Original entry on oeis.org

1, 2, 27, 1024, 9765625, 272097792, 558545864083284007, 295147905179352825856, 1144561273430837494885949696427, 305175781250000000000000000000000000, 1890591424712781041871514584574319778449301246603238034051, 98746073676238604311280222171685832518740805156864

Offset: 1

Ethan Ji, Jul 09 2025

nonn

Ethan Ji, Table of n, a(n) for n = 1..36

Cf. A076113.

a[n_Integer?Positive] := Module[{pairs = FactorInteger[n]}, Times @@ (#1^(n^2*(#2 #1^(2 #2) - (#1^#2 (#1^#2 - 1))/(#1 - 1))/(2 #1^(2 #2))) & @@@ pairs)]

a(n) = my(f=factor(n)); for (i=1, #f~, my(p=f[i,1], k=f[i,2]); f[i,2] = n^2*(k*p^(2*k) - p^k*(p^k-1)/(p-1))/(2*p^(2*k))); factorback(f); \\ Michel Marcus, Jul 10 2025

a(n) = Product_{p^k | n : prime p, k = p-adic order of n} p^(n^2*(k*p^(2k) - p^k(p^k - 1)/(p - 1)) / (2*p^(2k))).

a(p) = A076113(p), for prime p.

cp's OEIS Frontend

A002489 a(n) = n^(n^2), or (n^n)^n.

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A355400 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}.

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A090588 Number of labeled idempotent groupoids.

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A030257 Number of nonisomorphic commutative idempotent groupoids.

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A038017 Number of n-element commutative groupoids with an identity ("pointed" groupoids).

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A076112 Triangle (read by rows) in which the n-th row contains first n terms of n geometric progression with first term 1 and common ratio (n-1).

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A355561 Number of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= n^(i-1).

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A090599 Number of n-element labeled commutative groupoids with an identity.