A002109 -id:A002109 - cp's OEIS frontend

A000312 a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000, 285311670611, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, 18446744073709551616, 827240261886336764177, 39346408075296537575424, 1978419655660313589123979

Offset: 0

N. J. A. Sloane

Also number of labeled pointed rooted trees (or vertebrates) on n nodes.
For n >= 1 a(n) is also the number of n X n (0,1) matrices in which each row contains exactly one entry equal to 1. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
Also the number of labeled rooted trees on (n+1) nodes such that the root is lower than its children. Also the number of alternating labeled rooted ordered trees on (n+1) nodes such that the root is lower than its children. - Cedric Chauve (chauve(AT)lacim.uqam.ca), Mar 27 2002
With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j, i) = the j-th part of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i=1..p(n)} (n!/(Product_{j=1..p(i)} p(i, j)!)) * ((n!/(n - p(i)))!/(Product_{j=1..d(i)} m(i, j)!)). - Thomas Wieder, May 18 2005
All rational solutions to the equation x^y = y^x, with x < y, are given by x = A000169(n+1)/A000312(n), y = A000312(n+1)/A007778(n), where n = 1, 2, 3, ... . - Nick Hobson, Nov 30 2006
a(n) is the total number of leaves in all (n+1)^(n-1) trees on {0,1,2,...,n} rooted at 0. For example, with edges directed away from the root, the trees on {0,1,2} are {0->1,0->2},{0->1->2},{0->2->1} and contain a total of a(2)=4 leaves. - David Callan, Feb 01 2007
Limit_{n->infinity} A000169(n+1)/a(n) = exp(1). Convergence is slow, e.g., it takes n > 74 to get one decimal place correct and n > 163 to get two of them. - Alonso del Arte, Jun 20 2011
Also smallest k such that binomial(k, n) is divisible by n^(n-1), n > 0. - Michel Lagneau, Jul 29 2013
For n >= 2 a(n) is represented in base n as "one followed by n zeros". - R. J. Cano, Aug 22 2014
Number of length-n words over the alphabet of n letters. - Joerg Arndt, May 15 2015
Number of prime parking functions of length n+1. - Rui Duarte, Jul 27 2015
The probability density functions p(x, m=q, n=q, mu=1) = A000312(q)*E(x, q, q) and p(x, m=q, n=1, mu=q) = (A000312(q)/A000142(q-1))*x^(q-1)*E(x, q, 1), with q >= 1, lead to this sequence, see A163931, A274181 and A008276. - Johannes W. Meijer, Jun 17 2016
Satisfies Benford's law [Miller, 2015]. - N. J. A. Sloane, Feb 12 2017
A signed version of this sequence apart from the first term (1, -4, -27, 256, 3125, -46656, ...), has the following property: for every prime p == 1 (mod 2n), (-1)^(n(n-1)/2)*n^n = A057077(n)*a(n) is always a 2n-th power residue modulo p. - Jianing Song, Sep 05 2018
From Juhani Heino, May 07 2019: (Start)
n^n is both Sum_{i=0..n} binomial(n,i)*(n-1)^(n-i)
  and Sum_{i=0..n} binomial(n,i)*(n-1)^(n-i)*i.
The former is the familiar binomial distribution of a throw of n n-sided dice, according to how many times a required side appears, 0 to n. The latter is the same but each term is multiplied by its amount. This means that if the bank pays the player 1 token for each die that has the chosen side, it is always a fair game if the player pays 1 token to enter - neither bank nor player wins on average.
Examples:
2-sided dice (2 coins): 4 = 1 + 2 + 1 = 1*0 + 2*1 + 1*2 (0 omitted from now on);
3-sided dice (3 long triangular prisms): 27 = 8 + 12 + 6 + 1 = 12*1 + 6*2 + 1*3;
4-sided dice (4 long square prisms or 4 tetrahedrons): 256 = 81 + 108 + 54 + 12 + 1 = 108*1 + 54*2 + 12*3 + 1*4;
5-sided dice (5 long pentagonal prisms): 3125 = 1024 + 1280 + 640 + 160 + 20 + 1 = 1280*1 + 640*2 + 160*3 + 20*4 + 1*5;
6-sided dice (6 cubes): 46656 = 15625 + 18750 + 9375 + 2500 + 375 + 30 + 1 = 18750*1 + 9375*2 + 2500*3 + 375*4 + 30*5 + 1*6.
(End)
For each n >= 1 there is a graph on a(n) vertices whose largest independent set has size n and whose independent set sequence is constant (specifically, for each k=1,2,...,n, the graph has n^n independent sets of size k). There is no graph of smaller order with this property (Ball et al. 2019). - David Galvin, Jun 13 2019
For n >= 2 and 1 <= k <= n, a(n)*(n + 1)/4 + a(n)*(k - 1)*(n + 1 - k)/2*n is equal to the sum over all words w = w(1)...w(n) of length n over the alphabet {1, 2, ..., n} of the following quantity: Sum_{i=1..w(k)} w(i). Inspired by Problem 12432 in the AMM (see links). - Sela Fried, Dec 10 2023
Also, dimension of the unique cohomology group of the smallest interval containing the poset of partitions decorated by Perm, i.e. the poset of pointed partitions. - Bérénice Delcroix-Oger, Jun 25 2025

			G.f. = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + 3125*x^5 + 46656*x^6 + 823543*x^7 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 62, 63, 87.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 173, #39.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.37)
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).

Kenny Lau, Table of n, a(n) for n = 0..385 [First 100 terms computed by T. D. Noe]
Taylor Ball, David Galvin, Katie Hyry, and Kyle Weingartner, Independent set and matching permutations, arXiv:1901.06579 [math.CO], 2019.
Arthur T. Benjamin and Fritz Juhnke, Another way of counting n^n, SIAM J. Discrete Math., Vol. 5, No. 3 (1992), pp. 377-379. - _N. J. A. Sloane_, Jun 09 2011
H. Bottomley, Illustration of initial terms.
H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, The Mathematical Intelligencer, Vol. 20 (4), 1998, pp. 25-29. (Sequence appears as formula in Eq. (8))
C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 34.
Frank Ellermann, Illustration of binomial transforms.
José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, Bulletin of the Korean Mathematical Society, Vol. 51, No. 5 (2014), pp. 1325-1337; arXiv preprint, arXiv:1203.4066 [math.NT], 2012.
Nick Hobson, Solution to puzzle 48: Exponential equation.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 36.
Steven J. Miller (ed.), Exercises to "The Theory and Applications of Benford's Law", Princeton University Press, 2015.
Mustafa Obaid et al., The number of complete exceptional sequences for a Dynkin algebra, arXiv preprint arXiv:1307.7573 [math.RT], 2013.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
E. Vigren (Proposer), Problem 12432, Amer. Math. Monthly 130 (2023), p. 953.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Eric Weisstein's World of Mathematics, Hadamard's Maximum Determinant Problem.
Eric Weisstein's World of Mathematics, Hankel Matrix.
Dimitri Zvonkine, An algebra of power series..., arXiv:math/0403092 [math.AG], 2004.
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to Benford's law

Cf. A000107, A000169, A000272, A001372, A007778, A007830, A008785-A008791, A019538, A048993, A008279, A085741, A062206, A212333.
First column of triangle A055858. Row sums of A066324.
Cf. A001923 (partial sums), A002109 (partial products), A007781 (first differences), A066588 (sum of digits).
Cf. A056665, A081721, A130293, A168658, A275549-A275558 (various classes of endofunctions).
Cf. A174824, A204688.
Cf. A055137, A083648.

a000312 n = n ^ n
a000312_list = zipWith (^) [0..] [0..]  -- Reinhard Zumkeller, Jul 07 2012

A000312 := n->n^n: seq(A000312(n), n=0..17);

Array[ #^# &, 16] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Table[Sum[StirlingS2[n, i] i! Binomial[n, i], {i, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, Mar 17 2009 *)
a[ n_] := If[ n < 1, Boole[n == 0], n^n]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (1 + LambertW[-x]), {x, 0, n}]]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[n < 0, 0, n! SeriesCoefficient[ Nest[ 1 / (1 - x / (1 - Integrate[#, x])) &, 1 + O[x], n], {x, 0, n}]]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[ n < 0, 0, With[{m = n + 1}, m! SeriesCoefficient[ InverseSeries[ Series[ (x - 1) Log[1 - x], {x, 0, m}]], m]]]; (* Michael Somos, May 24 2014 *)

A000312[n]:=if n=0 then 1 else n^n$
makelist(A000312[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

PARI
```
{a(n) = n^n};
```

is(n)=my(b,k=ispower(n,,&b));if(k,for(e=1,valuation(k,b), if(k/b^e == e, return(1)))); n==1 \\ Charles R Greathouse IV, Jan 14 2013

{a(n) = my(A = 1 + O(x)); if( n<0, 0, for(k=1, n, A = 1 / (1 - x / (1 - intformal( A)))); n! * polcoeff( A, n))}; /* Michael Somos, May 24 2014 */

def A000312(n): return n**n # Chai Wah Wu, Nov 07 2022

a(n-1) = -Sum_{i=1..n} (-1)^i*i*n^(n-1-i)*binomial(n, i). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
E.g.f.: 1/(1 + W(-x)), W(x) = principal branch of Lambert's function.
a(n) = Sum_{k>=0} binomial(n, k)*Stirling2(n, k)*k! = Sum_{k>=0} A008279(n,k)*A048993(n,k) = Sum_{k>=0} A019538(n,k)*A007318(n,k). - Philippe Deléham, Dec 14 2003
E.g.f.: 1/(1 - T), where T = T(x) is Euler's tree function (see A000169).
a(n) = A000169(n+1)*A128433(n+1,1)/A128434(n+1,1). - Reinhard Zumkeller, Mar 03 2007
Comment on power series with denominators a(n): Let f(x) = 1 + Sum_{n>=1} x^n/n^n. Then as x -> infinity, f(x) ~ exp(x/e)*sqrt(2*Pi*x/e). - Philippe Flajolet, Sep 11 2008
E.g.f.: 1 - exp(W(-x)) with an offset of 1 where W(x) = principal branch of Lambert's function. - Vladimir Kruchinin, Sep 15 2010
a(n) = (n-1)*a(n-1) + Sum_{i=1..n} binomial(n, i)*a(i-1)*a(n-i). - Vladimir Shevelev, Sep 30 2010
With an offset of 1, the e.g.f. is the compositional inverse ((x - 1)*log(1 - x))^(-1) = x + x^2/2! + 4*x^3/3! + 27*x^4/4! + .... - Peter Bala, Dec 09 2011
a(n) = denominator((1 + 1/n)^n) for n > 0. - Jean-François Alcover, Jan 14 2013
a(n) = A089072(n,n) for n > 0. - Reinhard Zumkeller, Mar 18 2013
a(n) = (n-1)^(n-1)*(2*n) + Sum_{i=1..n-2} binomial(n, i)*(i^i*(n-i-1)^(n-i-1)), n > 1, a(0) = 1, a(1) = 1. - Vladimir Kruchinin, Nov 28 2014
log(a(n)) = lim_{k->infinity} k*(n^(1+1/k) - n). - Richard R. Forberg, Feb 04 2015
From Ilya Gutkovskiy, Jun 18 2016: (Start)
Sum_{n>=1} 1/a(n) = 1.291285997... = A073009.
Sum_{n>=1} 1/a(n)^2 = 1.063887103... = A086648.
Sum_{n>=1} n!/a(n) = 1.879853862... = A094082. (End)
A000169(n+1)/a(n) -> e, as n -> oo. - Daniel Suteu, Jul 23 2016
a(n) = n!*Product_{k=1..n} binomial(n, k)/Product_{k=1..n-1} binomial(n-1, k) = n!*A001142(n)/A001142(n-1). - Tony Foster III, Sep 05 2018
a(n-1) = abs(p_n(2-n)), for n > 2, the single local extremum of the n-th row polynomial of A055137 with Bagula's sign convention. - Tom Copeland, Nov 15 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = A083648. - Amiram Eldar, Jun 25 2021
Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = e (see Brothers/Knox link). - Harlan J. Brothers, Oct 24 2021
Conjecture: a(n) = Sum_{i=0..n} A048994(n, i) * A048993(n+i, n) for n >= 0; proved by Mike Earnest, see link at A354797. - Werner Schulte, Jun 19 2022

A074962 Decimal expansion of Glaisher-Kinkelin constant A.

Original entry on oeis.org

1, 2, 8, 2, 4, 2, 7, 1, 2, 9, 1, 0, 0, 6, 2, 2, 6, 3, 6, 8, 7, 5, 3, 4, 2, 5, 6, 8, 8, 6, 9, 7, 9, 1, 7, 2, 7, 7, 6, 7, 6, 8, 8, 9, 2, 7, 3, 2, 5, 0, 0, 1, 1, 9, 2, 0, 6, 3, 7, 4, 0, 0, 2, 1, 7, 4, 0, 4, 0, 6, 3, 0, 8, 8, 5, 8, 8, 2, 6, 4, 6, 1, 1, 2, 9, 7, 3, 6, 4, 9, 1, 9, 5, 8, 2, 0, 2, 3, 7, 4, 3, 9, 4, 2, 0, 6, 4, 6, 1, 2, 0

Offset: 1

Benoit Cloitre, Oct 05 2002

Arises in expressions such as A002109(n) = 1^1*2^2*3^3*...*n^n which is asymptotic to A*n^(n^2/2 + n/2 + 1/12)*exp(-n^2/4). See A002109 for more references and links.

Named after the English mathematician and astronomer James Whitbread Lee Glaisher (1848-1928) and the Swiss mathematician Hermann Kinkelin (1832-1913). - Amiram Eldar, Jun 15 2021

			1.2824271291006226368753425688697917277676889273250011920637400217404...

Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, p. 135.
Konrad Knopp, Theory and applications of infinite series, Dover, p. 555.

Gheorghe Coserea, Table of n, a(n) for n = 1..10010
L. Almodovar, V. H. Moll, H. Quand, Infinite products arising in paperfolding, JIS 19 (2016) # 16.5.1 eq. (13)
Chao-Ping Chen and Long Lin, Asymptotic expansions related to Glaisher-Kinkelin constant based on the Bell polynomials, Journal of Number Theory, Vol. 133 (2013), pp. 2699-2705.
Ovidiu Furdui, proposer, Problem 11494, Amer. Math. Monthly, Vol. 118, No. 9 (2011), 850-852.
J. W. L. Glaisher, On the Product 1^1.2^2.3^3...n^n, The Messenger of Mathematics, Vol. 7 (1878), pp. 43-47.
Antonio Gracia Llorente, A Simple Limit-Product Formula for Glaisher’s Constant, OSF Preprint, 2025.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J., Vol. 16 (2008), pp. 247-270; see Examples 5.2, 5.7, 5.11.
Fredrik Johansson et al., mpmath, Mathematical constants (Mpmath).
Fredrik Johansson et al., mpmath, Glaisher's constant to 20,000 digits.
Hermann Kinkelin, Über eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, Journal für die reine und angewandte Mathematik, Vol. 57 (1860), pp. 122-138.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., Vol. 332, No. 1 (2007), pp. 292-314; see Section 5.
Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.
Wikipedia, Glaisher-Kinkelin constant.

Cf. A001620, A243262, A243263, A243264, A243265.

Cf. A000178, A002109, A051675, A255321, A255323, A255344.

evalf(limit(product(k^k,k=1..n)/(n^(n^2/2+n/2+1/12)*exp(-n^2/4)),n=infinity),120); # Vaclav Kotesovec, Oct 23 2014

RealDigits[Glaisher, 10, 111][[1]] (* Robert G. Wilson v *)

x=10^(-100); exp(1/12-(zeta(-1+x)-zeta(-1))/x)

exp(1/12-zeta'(-1)) \\ Charles R Greathouse IV, Dec 12 2013

A = 2^(1/36)*Pi^(1/6)*exp(1/3*(-Gamma/4 + s(2)/3 - s(3)/4 + ...)) where s(k) denotes Sum_{n>=0} 1/(2n+1)^k.
Closed expressions for A are exp(-zeta'(2)/2/Pi^2 + log(2*Pi)/12 + Gamma/12) or exp(1/12-zeta'(-1)).
Equals (2*Pi)^(1/4) / limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2). - Vaclav Kotesovec, Dec 02 2023
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^4-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(2)/2 = 1/12 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
Equals e^(-1/4 + Integral_{x=1..2} x*log(sqrt(2*Pi)) - B_2(x) + x^2*Psi(x)/2 dx), where B_2(x) is the second Bernoulli polynomial and Psi(x) is the digamma function. - Andrea Pinos, Apr 16 2024
Equals exp(1/12 - 2*Integral_{x=0..oo} x*log(x)/(exp(2*Pi*x) - 1) dx) = exp(1/3 + 7*log(2)/36 - log(Pi)/6 + (2/3)*Integral_{x=0..1/2} log(Gamma(x+1)) dx) (see Finch). - Stefano Spezia, Dec 01 2024
From Antonio Graciá Llorente, May 03 2025: (Start)
Equals lim_{n->oo} (2^(13/3)*n)^(1/12) * Product_{k=1..n} (1 - 1/(2*k+1)^2)^((2*k+1)/6).
Equals lim_{n->oo} (24*n^2)^(1/24) * Product_{prime p<=n} (p^(1 - p/(p^2-1)) / sqrt(p^2-1))^(1/12). (End)

More terms from Sascha Kurz, Feb 03 2003

A000178 Superfactorials: product of first n factorials.

Original entry on oeis.org

1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000, 265790267296391946810949632000000000, 127313963299399416749559771247411200000000000, 792786697595796795607377086400871488552960000000000000

Offset: 0

N. J. A. Sloane

a(n) is also the Vandermonde determinant of the numbers 1,2,...,(n+1), i.e., the determinant of the (n+1) X (n+1) matrix A with A[i,j] = i^j, 1 <= i <= n+1, 0 <= j <= n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
a(n) = (1/n!) * D(n) where D(n) is the determinant of order n in which the (i,j)-th element is i^j. - Amarnath Murthy, Jan 02 2002
Determinant of S_n where S_n is the n X n matrix S_n(i,j) = Sum_{d|i} d^j. - Benoit Cloitre, May 19 2002
Appears to be det(M_n) where M_n is the n X n matrix with m(i,j) = J_j(i) where J_k(n) denote the Jordan function of row k, column n (cf. A059380(m)). - Benoit Cloitre, May 19 2002
a(2n+1) = 1, 12, 34560, 125411328000, ... is the Hankel transform of A000182 (tangent numbers) = 1, 2, 16, 272, 7936, ...; example: det([1, 2, 16, 272; 2, 16, 272, 7936; 16, 272, 7936, 353792; 272, 7936, 353792, 22368256]) = 125411328000. - Philippe Deléham, Mar 07 2004
Determinant of the (n+1) X (n+1) matrix whose i-th row consists of terms 1 to n+1 of the Lucas sequence U(i,Q), for any Q. When Q=0, the Vandermonde matrix is obtained. - T. D. Noe, Aug 21 2004
Determinant of the (n+1) X (n+1) matrix A whose elements are A(i,j) = B(i+j) for 0 <= i,j <= n, where B(k) is the k-th Bell number, A000110(k) [I. Mezo, JIS 14 (2011) # 11.1.1]. - T. D. Noe, Dec 04 2004
The Hankel transform of the sequence A090365 is A000178(n+1); example: det([1,1,3; 1,3,11; 3,11,47]) = 12. - Philippe Deléham, Mar 02 2005
Theorem 1.3, page 2, of Polynomial points, Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6, provides an example of an Abelian quotient group of order (n-1) superfactorial, for each positive integer n. The quotient is obtained from sequences of polynomial values. - E. F. Cornelius, Jr. (efcornelius(AT)comcast.net), Apr 09 2007
Starting with offset 1 this is a 'Matryoshka doll' sequence with alpha=1, the multiplicative counterpart to the additive A000292. seq(mul(mul(i,i=alpha..k), k=alpha..n),n=alpha..12). - Peter Luschny, Jul 14 2009
For n>0, a(n) is also the determinant of S_n where S_n is the n X n matrix, indexed from 1, S_n(i,j)=sigma_i(j), where sigma_k(n) is the generalized divisor sigma function: A000203 is sigma_1(n). - Enrique Pérez Herrero, Jun 21 2010
a(n) is the multiplicative Wiener index of the (n+1)-vertex path. Example: a(4)=288 because in the path on 5 vertices there are 3 distances equal to 2, 2 distances equal to 3, and 1 distance equal to 4 (2*2*2*3*3*4=288). See p. 115 of the Gutman et al. reference. - Emeric Deutsch, Sep 21 2011
a(n-1) = Product_{j=1..n-1} j! = V(n) = Product_{1 <= i < j <= n} (j - i) (a Vandermondian V(n), see the Ahmed Fares May 06 2001 comment above), n >= 1, is in fact the determinant of any n X n matrix M(n) with entries M(n;i,j) = p(j-1,x = i), 1 <= i, j <= n, where p(m,x), m >= 0, are monic polynomials of exact degree m with p(0,x) = 1. This is a special x[i] = i choice in a general theorem given in Vein-Dale, p. 59 (written for the transposed matrix M(n;j,x_i) = p(i-1,x_j) = P_i(x_j) in Vein-Dale, and there a_{k,k} = 1, for k=1..n). See the Aug 26 2013 comment under A049310, where p(n,x) = S(n,x) (Chebyshev S). - Wolfdieter Lang, Aug 27 2013
a(n) is the number of monotonic magmas on n elements labeled 1..n with a symmetric multiplication table. I.e., Product(i,j) >= max(i,j); Product(i,j) = Product(j,i). - Chad Brewbaker, Nov 03 2013
The product of the pairwise differences of n+1 integers is a multiple of a(n) [and this does not hold for any k > a(n)]. - Charles R Greathouse IV, Aug 15 2014
a(n) is the determinant of the (n+1) X (n+1) matrix M with M(i,j) = (n+j-1)!/(n+j-i)!, 1 <= i <= n+1, 1 <= j <= n+1. - Stoyan Apostolov, Aug 26 2014
All terms are in A064807 and all terms after a(2) are in A005101. - Ivan N. Ianakiev, Sep 02 2016
Empirical: a(n-1) is the determinant of order n in which the (i,j)-th entry is the (j-1)-th derivative of x^(x+i-1) evaluated at x=1. - John M. Campbell, Dec 13 2016
Empirical: If f(x) is a smooth, real-valued function on an open neighborhood of 0 such that f(0)=1, then a(n) is the determinant of order n+1 in which the (i,j)-th entry is the (j-1)-th derivative of f(x)/((1-x)^(i-1)) evaluated at x=0. - John M. Campbell, Dec 27 2016
Also the automorphism group order of the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Is the zigzag Hankel transform of A000182. That is, a(2*n+1) is the Hankel transform of A000182 and a(2*n+2) is the Hankel transform of A000182(n+1). - Michael Somos, Mar 11 2020
Except for n = 0, 1, superfactorial a(n) is never a square (proof in link Mabry and Cormick, FFF 4 p. 349); however, when k belongs to A349079 (see for further information), there exists m, 1 <= m <= k such that a(k) / m! is a square. - Bernard Schott, Nov 29 2021

			a(3) = (1/6)* | 1 1 1 | 2 4 8 | 3 9 27 |
a(7) = n! * a(n-1) = 7! * 24883200 = 125411328000.
a(12) = 1! * 2! * 3! * 4! * 5! * 6! * 7! * 8! * 9! * 10! * 11! * 12!
= 1^12 * 2^11 * 3^10 * 4^9 * 5^8 * 6^7 * 7^6 * 8^5 * 9^4 * 10^3 * 11^2 * 12^1
= 2^56 * 3^26 * 5^11 * 7^6 * 11^2.
G.f. = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 231.
H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.
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).
R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.

Boris Hostnik, Table of n, a(n) for n = 0..46
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
Andreas B. G. Blobel, On convolution powers of 1/x, arXiv:2203.09519 [math.CO], 2022.
E. F. Cornelius, Jr. and Phill Schultz, Polynomial points , Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.
Selden Crary, Factorization of the Determinant of the Gaussian-Covariance Matrix of Evenly Spaced Points Using an Inter-dimensional Multiset Duality, arXiv preprint arXiv:1406.6326 [math.ST], 2014-2019.
N. Destainville, R. Mosseri and F. Bailly, Configurational Entropy of Codimension-One Tilings and Directed Membranes, J. Stat. Phys. 87, Nos 3/4, 697 (1997).
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
Richard Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107 (2000), 557-560.
William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 7.
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [Broken link]
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Ivan Gutman, Wolfgang Linert, István Lukovits and Željko Tomović, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., Vol. 40, No. 1 (2000), pp. 113-116.
Brady Haran and Sophie Maclean, What's special about 288?, Numberphile video (2023).
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Nick Hobson, Python program for this sequence.
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
Pavel L. Krapivsky, Jean-Marc Luck and Kirone Mallick, Quantum return probability of a system of N non-interacting lattice fermions, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 2 (2018), 023104; arXiv preprint, arXiv:1710.08178 [cond-mat.mes-hall], 2017-2018.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; arXiv preprint, arXiv:1409.4145 [math.NT], 2014-2015.
Mogens Esrom Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009.
Rémy Mosseri and Francis Bailly, Configurational Entropy in Octagonal Tiling Models, Int. J. Mod. Phys. B, Vol. 7, No. 6-7 (1993), pp. 1427-1436.
Rémy Mosseri, F. Bailly and C. Sire, Configurational Entropy in Random Tiling Models, J. Non-Cryst. Solids, Vol. 153-154 (1993), pp. 201-204.
Amarnath Murthy, Miscellaneous Results and Theorems on Smarandache terms and factor partitions, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 3.14.
Christian Radoux, Query 145, Notices Amer. Math. Soc., 25-3 (1978), p. 197.
Christian Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
Vignesh Raman, The Generalized Superfactorial, Hyperfactorial and Primorial functions, arXiv:2012.00882 [math.NT], 2020.
Michel Waldschmidt, Schanuel Property for Elliptic and Quasi-Elliptic Functions, arXiv:2504.14041 [math.NT], 2025. Mentions this sequence, see p. 10.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Eric Weisstein's World of Mathematics, Bell Number.
Eric Weisstein's World of Mathematics, Factorial Products.
Eric Weisstein's World of Mathematics, Graph Automorphism.
Eric Weisstein's World of Mathematics, Lucas Sequence.
Eric Weisstein's World of Mathematics, Superfactorial.
Eric Weisstein's World of Mathematics, Vandermonde Determinant.
Index to divisibility sequences.
Index entries for sequences related to factorial numbers.
Index to sequences related to Olympiads and other Mathematical competitions.

Partial products of A000142.
Cf. A002109, A036561, A000292, A098694, A098695, A113271, A087316, A113208, A113231, A113257, A113258, A113320, A113336, A113498, A113173, A113170, A113475, A113492, A113497, A113533, A113534, A113535, A113153, A113154, A113122, A114045, A055462, A137986, A137987.
A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
A000178 is the Hankel transform (see A001906 for definition) of A000085, A000110, A000296, A005425, A005493, A005494 and A045379. - John W. Layman, Jul 28 2000
Cf. A255322, A255358, A255359, A255360.
Cf. A051675, A255321, A255323, A255344, A287013.
Cf. A348692, A349079.

[&*[Factorial(k): k in [0..n]]: n in [0..20]]; // Bruno Berselli, Mar 11 2015

A000178 := proc(n)
    mul(i!,i=1..n) ;
end proc:
seq(A000178(n),n=0..10) ; # R. J. Mathar, Oct 30 2015

a[0] := 1; a[1] := 1; a[n_] := n!*a[n - 1]; Table[a[n], {n, 1, 12}] (* Stefan Steinerberger, Mar 10 2006 *)
Table[BarnesG[n], {n, 2, 14}] (* Zerinvary Lajos, Jul 16 2009 *)
FoldList[Times,1,Range[20]!] (* Harvey P. Dale, Mar 25 2011 *)
RecurrenceTable[{a[n] == n! a[n - 1], a[0] == 1}, a, {n, 0, 12}] (* Ray Chandler, Jul 30 2015 *)
BarnesG[Range[2, 20]] (* Eric W. Weisstein, Jul 14 2017 *)

A000178(n):=prod(k!,k,0,n)$ makelist(A000178(n),n,0,30); /* Martin Ettl, Oct 23 2012 */

A000178(n)=prod(k=2,n,k!) \\ M. F. Hasler, Sep 02 2007

a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, (1+j!*x+x*O(x^n)) )), n) \\ Paul D. Hanna, Oct 02 2013

for(j=1,13, print1(prod(k=1,j,k^(j-k)),", ")) \\ Hugo Pfoertner, Apr 09 2020

A000178_list, n, m = [1], 1,1
for i in range(1,100):
    m *= i
    n *= m
    A000178_list.append(n) # Chai Wah Wu, Aug 21 2015

from math import prod
def A000178(n): return prod(i**(n-i+1) for i in range(2,n+1)) # Chai Wah Wu, Nov 26 2023

def mono_choices(a,b,n)
    n - [a,b].max
end
def comm_mono_choices(n)
    accum =1
    0.upto(n-1) do |i|
        i.upto(n-1) do |j|
            accum = accum * mono_choices(i,j,n)
        end
    end
    accum
end
1.upto(12) do |k|
    puts comm_mono_choices(k)
end # Chad Brewbaker, Nov 03 2013

a(0) = 1, a(n) = n!*a(n-1). - Lee Hae-hwang, May 13 2003, corrected by Ilya Gutkovskiy, Jul 30 2016
a(0) = 1, a(n) = 1^n * 2^(n-1) * 3^(n-2) * ... * n = Product_{r=1..n} r^(n-r+1). - Amarnath Murthy, Dec 12 2003 [Formula corrected by Derek Orr, Jul 27 2014]
a(n) = sqrt(A055209(n)). - Philippe Deléham, Mar 07 2004
a(n) = Product_{i=1..n} Product_{j=0..i-1} (i-j). - Paul Barry, Aug 02 2008
log a(n) = 0.5*n^2*log n - 0.75*n^2 + O(n*log n). - Charles R Greathouse IV, Jan 13 2012
Asymptotic: a(n) ~ exp(zeta'(-1) - 3/4 - (3/4)*n^2 - (3/2)*n)*(2*Pi)^(1/2 + (1/2)*n)*(n+1)^((1/2)*n^2 + n + 5/12). For example, a(100) is approx. 0.270317...*10^6941. (See A213080.) - Peter Luschny, Jun 23 2012
G.f.: 1 + x/(U(0) - x) where U(k) = 1 + x*(k+1)! - x*(k+2)!/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2012
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 + 1/((k+1)!*x*G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k!*x). - Paul D. Hanna, Oct 02 2013
A203227(n+1)/a(n) -> e, as n -> oo. - Daniel Suteu, Jul 30 2016
From Ilya Gutkovskiy, Jul 30 2016: (Start)
a(n) = G(n+2), where G(n) is the Barnes G-function.
a(n) ~ exp(1/12 - n*(3*n+4)/4)*n^(n*(n+2)/2 + 5/12)*(2*Pi)^((n+1)/2)/A, where A is the Glaisher-Kinkelin constant (A074962).
Sum_{n>=0} (-1)^n/a(n) = A137986. (End)
0 = a(n)*a(n+2)^3 + a(n+1)^2*a(n+2)^2 - a(n+1)^3*a(n+3) for all n in Z (if a(-1)=1). - Michael Somos, Mar 11 2020
Sum_{n>=0} 1/a(n) = A287013 = 1/A137987. - Amiram Eldar, Nov 19 2020
a(n) = Wronskian(1, x, x^2, ..., x^n). - Mohammed Yaseen, Aug 01 2023
From Andrea Pinos, Apr 04 2024: (Start)
a(n) = e^(Sum_{k=1..n} (Integral_{x=1..k+1} Psi(x) dx)).
a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + x*Psi(x)) dx).
a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + (n+1)*Psi(x) - log(Gamma(x))) dx).
Psi(x) is the digamma function. (End)

A002064 Cullen numbers: a(n) = n*2^n + 1.

Original entry on oeis.org

1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521, 44040193, 92274689, 192937985, 402653185, 838860801, 1744830465, 3623878657, 7516192769, 15569256449, 32212254721, 66571993089

Offset: 0

N. J. A. Sloane

Binomial transform is A084859. Inverse binomial transform is A004277. - Paul Barry, Jun 12 2003
Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1] =-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 26 2010
Indices of primes are listed in A005849. - M. F. Hasler, Jan 18 2015
Add the list of fractions beginning with 1/2 + 3/4 + 7/8 + ... + (2^n - 1)/2^n and take the sums pairwise from left to right. For 1/2 + 3/4 = 5/4, 5 + 4 = 9 = a(2); for 5/4 + 7/8 = 17/8, 17 + 8 = 25 = a(3); for 17/8 + 15/16 = 49/16, 49 + 16 = 65 = a(4); for 49/16 + 31/32 = 129/32, 129 + 32 = 161 = a(5). For each pairwise sum a/b, a + b = n*2^(n+1). - J. M. Bergot, May 06 2015
Number of divisors of (2^n)^(2^n). - Gus Wiseman, May 03 2021
Named after the Irish Jesuit priest James Cullen (1867-1933), who checked the primality of the terms up to n=100. - Amiram Eldar, Jun 05 2021

			G.f. = 1 + 3*x + 9*x^2 + 25*x^3 + 65*x^4 + 161*x^5 + 385*x^6 + 897*x^7 + ... - _Michael Somos_, Jul 18 2018

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
R. K. Guy, Unsolved Problems in Number Theory, B20.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 240-242.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 346.
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).

T. D. Noe, Table of n, a(n) for n=0..300
Ray Ballinger, Cullen Primes: Definition and Status.
Attila Bérczes, István Pink, and Paul Thomas Young, Cullen numbers and Woodall numbers in generalized Fibonacci sequences, J. Num. Theor. (2024) Vol. 262, 86-102.
Yuri Bilu, Diego Marques, and Alain Togbé, Generalized Cullen numbers in linear recurrence sequences, Journal of Number Theory, Vol. 202 (2019), pp. 412-425; arXiv preprint, arXiv:1806.09441 [math.NT], 2018.
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
C. K. Caldwell, The Top Twenty: Cullen Primes.
James Cullen, Question 15897, Educational Times, Vol. 58 (December 1905), p. 534.
Orhan Eren and Yüksel Soykan, Gaussian Generalized Woodall Numbers, Arch. Current Res. Int'l (2023) Vol. 23, Iss. 8, Art. No. ACRI.108618, 48-68. See p. 50.
Orhan Eren and Yüksel Soykan, On Dual Hyperbolic Generalized Woodall Numbers, Archives Current Res. Int'l (2024) Vol. 24, Iss. 11, Art. No. ACRI.126420, 398-423. See p. 401.
Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020.
José María Grau and Florian Luca, Cullen numbers with the Lehmer property, Proceedings of the American Mathematical Society, Vol. 140, No. 1 (2012), pp. 129-134; arXiv preprint, arXiv:1103.3578 [math.NT], Mar 18 2011.
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Diego Marques, On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.4.
Hisanori Mishima, Factorizations of many number sequences, Cullen numbers (n = 1 to 100), (n = 101 to 200), (n = 201 to 300), (n = 301 to 323).
Simon Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences, Vol. 8, No. 4 (2019), pp. 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences, Vol. 8, No. 10 (2019).
Eric Weisstein's World of Mathematics, Cullen Number.
Wikipedia, Cullen number.
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A005849, A003261, A050914, A130197, A134081, A001787, A143038, A156708, A181527.
Cf. A000325, A186947.
Diagonal k = n + 1 of A046688.
A000005 counts divisors of n.
A000312 = n^n.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
A173339 lists positions of squares in A062319.
A188385 gives the highest prime exponent in n^n.
A249784 counts divisors of n^n^n.
Cf. A000169, A000272, A036289, A066959, A176029, A340841, A343656.

a002064 n = n * 2 ^ n + 1
a002064_list = 1 : 3 : (map (+ 1) $ zipWith (-) (tail xs) xs)
   where xs = map (* 4) a002064_list
-- Reinhard Zumkeller, Mar 16 2013

[n*2^n + 1: n in [0..40]]; // Vincenzo Librandi, May 07 2015

A002064:=-(1-2*z+2*z**2)/((z-1)*(-1+2*z)**2); # conjectured by Simon Plouffe in his 1992 dissertation

Table[n*2^n+1,{n,0,2*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)
LinearRecurrence[{5,-8,4},{1,3,9},51] (* Harvey P. Dale, Oct 13 2011 *)
CoefficientList[Series[(1 - 2 x + 2 x^2)/((1 - x) (2 x - 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)

A002064(n)=n*2^n+1  \\ M. F. Hasler, Oct 31 2012

a(n) = 4a(n-1) - 4a(n-2) + 1. - Paul Barry, Jun 12 2003
a(n) = sum of row (n+1) of triangle A130197. Example: a(3) = 25 = (12 + 8 + 4 + 1), row 4 of A130197. - Gary W. Adamson, May 16 2007
Row sums of triangle A134081. - Gary W. Adamson, Oct 07 2007
Equals row sums of triangle A143038. - Gary W. Adamson, Jul 18 2008
Equals row sums of triangle A156708. - Gary W. Adamson, Feb 13 2009
G.f.: -(1-2*x+2*x^2)/((-1+x)*(2*x-1)^2). a(n) = A001787(n+1)+1-A000079(n). - R. J. Mathar, Nov 16 2007
a(n) = 1 + 2^(n + log_2(n)) ~ 1 + A000079(n+A004257(n)). a(n) ~ A000051(n+A004257(n)). - Jonathan Vos Post, Jul 20 2008
a(0)=1, a(1)=3, a(2)=9, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Harvey P. Dale, Oct 13 2011
a(n) = A036289(n) + 1 = A003261(n) + 2. - Reinhard Zumkeller, Mar 16 2013
E.g.f.: 2*x*exp(2*x) + exp(x). - Robert Israel, Dec 12 2014
a(n) = 2^n * A000325(n) = 4^n * A186947(-n) for all n in Z. - Michael Somos, Jul 18 2018
a(n) = Sum_{i=0..n-1} a(i) + A000325(n+1). - Ivan N. Ianakiev, Aug 07 2019
a(n) = sigma((2^n)^(2^n)) = A000005(A057156(n)) = A062319(2^n). - Gus Wiseman, May 03 2021
Sum_{n>=0} 1/a(n) = A340841. - Amiram Eldar, Jun 05 2021

Edited by M. F. Hasler, Oct 31 2012

A036740 a(n) = (n!)^n.

Original entry on oeis.org

1, 1, 4, 216, 331776, 24883200000, 139314069504000000, 82606411253903523840000000, 6984964247141514123629140377600000000, 109110688415571316480344899355894085582848000000000, 395940866122425193243875570782668457763038822400000000000000000000

Offset: 0

N. J. A. Sloane

(-1)^n*a(n) is the determinant of the n X n matrix m_{i,j} = T(n+i,j), 1 <= i,j <= n, where T(n,k) are the signed Stirling numbers of the first kind (A008275). Derived from methods given in Krattenthaler link. - Benoit Cloitre, Sep 17 2005
a(n) is also the number of binary operations on an n-element set which are right (or left) cancellative. These are also called right (left) cancellative magma or groupoids. The multiplication table of a right (left) cancellative magma is an n X n matrix with entries from an n element set such that the elements in each column (or row) are distinct. - W. Edwin Clark, Apr 09 2009
This sequence is mentioned in "Experimentation in Mathematics" as a sum-of-powers determinant. - John M. Campbell, May 07 2011
Determinant of the n X n matrix M_n = [m_n(i,j)] with m_n(i,j) = Stirling2(n+i,j) for 1<=i,j<=n. - Alois P. Heinz, Jul 26 2013

Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., 2004, p. 207.

Alois P. Heinz, Table of n, a(n) for n = 0..30
Christian Krattenthaler, Advanced determinant calculus, in: D. Foata and G. N. Han (eds.), The Andrews Festschrift, Springer, Berlin, Heidelberg, 2001, pp. 349-426; arXiv preprint, arXiv:math/9902004 [math.CO], 1999.

Cf. A000142, A000169, A000312.
Cf. A086687, A225764, A261114, A261490.
Main diagonal of A225816.

a:= n-> n!^n:
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 25 2013

Table[(n!)^n,{n,0,10}] (* Harvey P. Dale, Sep 29 2013 *)

makelist(n!^n,n,0,10); /* Martin Ettl, Jan 13 2013 */

PARI
```
a(n)=n!^n;
```

a(n) = a(n-1)*n^n*(n-1)! = a(n-1)*A000169(n)*A000142(n) = A036740(n-1) * A000312(n)*A000142(n-1). - Henry Bottomley, Dec 06 2001
From Benoit Cloitre, Sep 17 2005: (Start)
a(n) = Product_{k=1..n} (k-1)!*k^k;
a(n) = A000178(n-1)*A002109(n) for n >= 1. (End)
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12). - Vaclav Kotesovec, Nov 14 2014
a(n) = Product_{k=1..n} k^n. - José de Jesús Camacho Medina, Jul 12 2016
Sum_{n>=0} 1/a(n) = A261114. - Amiram Eldar, Nov 16 2020

A001142 a(n) = Product_{k=1..n} k^(2k - 1 - n).

Original entry on oeis.org

1, 1, 2, 9, 96, 2500, 162000, 26471025, 11014635520, 11759522374656, 32406091200000000, 231627686043080250000, 4311500661703860387840000, 209706417310526095716965894400, 26729809777664965932590782608648192

Offset: 0

N. J. A. Sloane

Absolute value of determinant of triangular matrix containing binomial coefficients.
These are also the products of consecutive horizontal rows of the Pascal triangle. - Jeremy Hehn (ROBO_HEN5000(AT)rose.net), Mar 29 2007
Limit_{n->oo} a(n)*a(n+2)/a(n+1)^2 = e, as follows from lim_{n->oo} (1 + 1/n)^n = e. - Harlan J. Brothers, Nov 26 2009
A000225 gives the positions of odd terms. - Antti Karttunen, Nov 02 2014
Product of all unreduced fractions h/k with 1 <= k <= h <= n. - Jonathan Sondow, Aug 06 2015
a(n) is a product of the binomial coefficients from the n-th row of the Pascal triangle, for n= 0, 1, 2, ... For n > 0, a(n) means the number of all maximum chains in the poset formed by the n-dimensional Boolean cube {0,1}^n and the relation "precedes by weight". This relation is defined over {0,1}^n as follows: for arbitrary vectors u, v of {0,1}^n we say that "u precedes by weight v" if wt(u) < wt(v) or if u = v, where wt(u) denotes the (Hamming) weight of u. For more details, see the sequence A051459. - Valentin Bakoev, May 17 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
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).

T. D. Noe, Table of n, a(n) for n = 0..50
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
Harlan J. Brothers, Finding e in Pascal's Triangle, Mathematics Magazine, 85 (2012), p. 51.
Harlan J. Brothers, Pascal's Triangle: The Hidden Stor-e, The Mathematical Gazette, 96 (2012), 145-148.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.
Leroy Quet, Problem 1636, Mathematics Magazine, Dec. 2001, p. 403.

Cf. A000178, A002109, A007318, A000225, A056077, A249421, A187059 (2-adic valuation), A249343, A249345, A249346, A249347, A249151.

Cf. also A004788, A056606 (squarefree kernel), A256113.

List([0..15], n-> Product([0..n], k-> Binomial(n,k) )); # G. C. Greubel, May 23 2019

a001142 = product . a007318_row -- Reinhard Zumkeller, Mar 16 2015

[(&*[Binomial(n,k): k in [0..n]]): n in [0..15]]; // G. C. Greubel, May 23 2019

a:=n->mul(binomial(n,k), k=0..n): seq(a(n), n=0..14); # Zerinvary Lajos, Jan 22 2008

Table[Product[k^(2*k - 1 - n), {k, n}], {n, 0, 20}] (* Harlan J. Brothers, Nov 26 2009 *)
Table[Hyperfactorial[n]/BarnesG[n+2], {n, 0, 20}] (* Peter Luschny, Nov 29 2015 *)
Table[Product[(n - k + 1)^(n - 2 k + 1), {k, 1, n}], {n, 0, 20}] (* Harlan J. Brothers, Aug 26 2023 *)

a(n):= prod(binomial(n,k),k,0,n); n : 15; for i from 0 thru n do print (a(i)); /* Valentin Bakoev, May 17 2019 */

for(n=0,16,print(prod(m=1,n,binomial(n,m))))

A001142(n) = prod(k=1, n, k^((k+k)-1-n)); \\ Antti Karttunen, Nov 02 2014

from math import factorial, prod
from fractions import Fraction
def A001142(n): return prod(Fraction((k+1)**k,factorial(k)) for k in range(1,n)) # Chai Wah Wu, Jul 15 2022

a = lambda n: prod(k^k/factorial(k) for k in (1..n))
[a(n) for n in range(20)] # Peter Luschny, Nov 29 2015

(define (A001142 n) (mul (lambda (k) (expt k (+ k k -1 (- n)))) 1 n))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
;; Antti Karttunen, Oct 28 2014

a(n) = C(n, 0)*C(n, 1)* ... *C(n, n).
From Harlan J. Brothers, Nov 26 2009: (Start)
a(n) = Product_{j=1..n-2} Product_{k=1..j} (1+1/k)^k, n >= 3.
a(1) = a(2) = 1, a(n) = a(n-1) * Product_{k=1..n-2} (1+1/k)^k. (End)
a(n) = hyperfactorial(n)/superfactorial(n) =  A002109(n)/A000178(n). - Peter Luschny, Jun 24 2012
a(n) ~ A^2 * exp(n^2/2 + n - 1/12) / (n^(n/2 + 1/3) * (2*Pi)^((n+1)/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015
a(n) = Product_{i=1..n} Product_{j=1..i} (i/j). - Pedro Caceres, Apr 06 2019
a(n) = Product_{k=1..n} (n - k + 1)^(n - 2*k + 1). - Harlan J. Brothers, Aug 26 2023

More terms from James Sellers, May 01 2000

Better description from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001

A001923 a(n) = Sum_{k=1..n} k^k.

Original entry on oeis.org

0, 1, 5, 32, 288, 3413, 50069, 873612, 17650828, 405071317, 10405071317, 295716741928, 9211817190184, 312086923782437, 11424093749340453, 449317984130199828, 18896062057839751444, 846136323944176515621

Offset: 0

N. J. A. Sloane

Starting from the second term, 1, the terms could be described as the special case (n=1; j=1) of the following general formula: a(n) =  Sum [(n + k - 1)]^(k) n=1; j=1; i=1,2,3,...,... For (n=0; j=1) the formula yields A062815 n=0; j=1; i=2,3,4,... For (n=2; j=0) we get A060946 and for (n=3; j=0) A117887. - Alexander R. Povolotsky, Sep 01 2007
From Luan Alberto Ferreira, Aug 01 2017: (Start)
If n == 0 or 3 (mod 4), then a(n) == 0 (mod 4).
If n == 0, 4, 7, 14, 15 or 17 (mod 18), then a(n) == 0 (mod 3). (End)
Called the hypertriangular function by M. K. Azarian. - Light Ediand, Nov 19 2021

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, p. 308.
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).

T. D. Noe, Table of n, a(n) for n=0..100
Mohammad K. Azarian, On the hyperfactorial function, hypertriangular function and the discriminants of certain polynomials, Int. J. Pure Appl. Math., Vol. 36, No. 2 (2007), pp. 251-257.
Andrew Cusumano, Problem H-656, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 45, No. 2 (2007), p. 187; A Sequence Tending To e, Solution to Problem H-656, ibid., Vol. 46-47, No. 3 (2008/2009), pp. 285-287.
G. W. Wishard (proposer) and F. Underwood (solution), Problem 4155: Bound for a Finite Sum, Amer. Math. Monthly, Vol. 53, No. 8 (1946), pp. 471-473.

Cf. A073825, A062970 (another version).
Cf. A062815, A060946, A117887.
Cf. A000312, A002109.

a001923 n = a001923_list !! n
a001923_list = scanl (+) 0 $ tail a000312_list
-- Reinhard Zumkeller, Jul 11 2014

Accumulate[Join[{0},Table[k^k,{k,20}]]] (* Harvey P. Dale, Feb 11 2015 *)

for(n=1,20,print1(sum(x=1,n,x^x), ", ")) \\ Jorge Coveiro, Dec 24 2004

# generates initial segment of sequence
from itertools import accumulate
def f(k): return 0 if k == 0 else k**k
def aupton(nn): return list(accumulate(f(k) for k in range(nn+1)))
print(aupton(17)) # Michael S. Branicky, Feb 12 2022

a(n) = A062970(n) - 1.
a(n+1)/a(n) > e*n and a(n+1)/a(n) is asymptotic to e*n. - Benoit Cloitre, Sep 29 2002
For n > 0: a(n) = a(n-1) + A000312(n). - Reinhard Zumkeller, Jul 11 2014
Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - Amiram Eldar, Jan 03 2022

A062319 Number of divisors of n^n, or of A000312(n).

Original entry on oeis.org

1, 1, 3, 4, 9, 6, 49, 8, 25, 19, 121, 12, 325, 14, 225, 256, 65, 18, 703, 20, 861, 484, 529, 24, 1825, 51, 729, 82, 1653, 30, 29791, 32, 161, 1156, 1225, 1296, 5329, 38, 1521, 1600, 4961, 42, 79507, 44, 4005, 4186, 2209, 48, 9457, 99, 5151, 2704, 5565, 54

Offset: 0

Jason Earls, Jul 05 2001

From Gus Wiseman, May 02 2021: (Start)
Conjecture: The number of divisors of n^n equals the number of pairwise coprime ordered n-tuples of divisors of n. Confirmed up to n = 30. For example, the a(1) = 1 through a(5) = 6 tuples are:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)  (1,1,1,1,1)
       (1,2)  (1,1,3)  (1,1,1,2)  (1,1,1,1,5)
       (2,1)  (1,3,1)  (1,1,1,4)  (1,1,1,5,1)
              (3,1,1)  (1,1,2,1)  (1,1,5,1,1)
                       (1,1,4,1)  (1,5,1,1,1)
                       (1,2,1,1)  (5,1,1,1,1)
                       (1,4,1,1)
                       (2,1,1,1)
                       (4,1,1,1)
The unordered case (pairwise coprime n-multisets of divisors of n) is counted by A343654.
(End)

			From _Gus Wiseman_, May 02 2021: (Start)
The a(1) = 1 through a(5) = 6 divisors:
  1  1  1   1    1
     2  3   2    5
     4  9   4    25
        27  8    125
            16   625
            32   3125
            64
            128
            256
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Cf. A046798, A077592, A035116. - Enrique Pérez Herrero, Nov 09 2010
Number of divisors of A000312(n).
Taking Omega instead of sigma gives A066959.
Positions of squares are A173339.
Diagonal n = k of the array A343656.
A000005 counts divisors.
A059481 counts k-multisets of elements of {1..n}.
A334997 counts length-k strict chains of divisors of n.
A343658 counts k-multisets of divisors.
Pairwise coprimality:
- A018892 counts coprime pairs of divisors.
- A084422 counts pairwise coprime subsets of {1..n}.
- A100565 counts pairwise coprime triples of divisors.
- A225520 counts pairwise coprime sets of divisors.
- A343652 counts maximal pairwise coprime sets of divisors.
- A343653 counts pairwise coprime non-singleton sets of divisors > 1.
- A343654 counts pairwise coprime sets of divisors > 1.
Cf. A000169, A000272, A002064, A002109, A009998, A048691, A143773, A146291, A176029, A327527, A343657.

[NumberOfDivisors(n^n): n in  [0..60]]; // Vincenzo Librandi, Nov 09 2014

A062319[n_IntegerQ]:=DivisorSigma[0,n^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
Join[{1},DivisorSigma[0,#^#]&/@Range[60]] (* Harvey P. Dale, Jun 06 2024 *)

je=[]; for(n=0,200,je=concat(je,numdiv(n^n))); je

{ for (n=0, 1000, write("b062319.txt", n, " ", numdiv(n^n)); ) } \\ Harry J. Smith, Aug 04 2009

a(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],fm[k,2]*n+1) \\ Franklin T. Adams-Watters, May 03 2011

a(n) = if(n==0, 1, sumdiv(n, d, n^omega(d))); \\ Seiichi Manyama, May 12 2021

from math import prod
from sympy import factorint
def A062319(n): return prod(n*d+1 for d in factorint(n).values()) # Chai Wah Wu, Jun 03 2021

a(n) = A000005(A000312(n)). - Enrique Pérez Herrero, Nov 09 2010
a(2^n) = A002064(n). - Gus Wiseman, May 02 2021
a(prime(n)) = prime(n) + 1. - Gus Wiseman, May 02 2021
a(n) = Product_{i=1..s} (1 + n * m_i) where (m_1,...,m_s) is the sequence of prime multiplicities (prime signature) of n. - Gus Wiseman, May 02 2021
a(n) = Sum_{d|n} n^omega(d) for n > 0. - Seiichi Manyama May 12 2021

A009998 Triangle in which j-th entry in i-th row is (j+1)^(i-j).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Read as a square array this is the Hilbert transform of triangle A123125 (see A145905 for the definition of this term). For example, the fourth row of A123125 is (0,1,4,1) and the expansion (x + 4*x^2 + x^3)/(1-x)^4 = x + 8*x^2 + 27*x^3 + 64*x^4 + ... generates the entries in the fourth row of this array read as a square. - Peter Bala, Oct 28 2008

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  8,  9,  4,  1;
  1, 16, 27, 16,  5,  1;
  1, 32, 81, 64, 25,  6,  1;
  ...
From _Gus Wiseman_, May 01 2021: (Start)
The rows of the triangle are obtained by reading antidiagonals upward in the following table of A(k,n) = n^k, with offset k = 0, n = 1:
         n=1:     n=2:     n=3:     n=4:     n=5:     n=6:
   k=0:   1        1        1        1        1        1
   k=1:   1        2        3        4        5        6
   k=2:   1        4        9       16       25       36
   k=3:   1        8       27       64      125      216
   k=4:   1       16       81      256      625     1296
   k=5:   1       32      243     1024     3125     7776
   k=6:   1       64      729     4096    15625    46656
   k=7:   1      128     2187    16384    78125   279936
   k=8:   1      256     6561    65536   390625  1679616
   k=9:   1      512    19683   262144  1953125 10077696
  k=10:   1     1024    59049  1048576  9765625 60466176
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 24.

T. D. Noe, Rows n=0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter Luschny, Figurate number - a very short introduction

Row sums give A026898.
Cf. A088956, A123125, A179927.
Column n = 2 of the array is A000079.
Column n = 3 of the array is A000244.
Row k = 2 of the array is A000290.
Row k = 3 of the array is A000578.
Diagonal n = k of the array is A000312.
Diagonal n = k + 1 of the array is A000169.
Diagonal n = k + 2 of the array is A000272.
The transpose of the array is A009999.
The numbers of divisors of the entries are A343656 (row sums: A343657).
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
Cf. A002109, A062319, A066959, A143773, A176029, A204688, A326358, A327527.

a009998 n k = (k + 1) ^ (n - k)
a009998_row n = a009998_tabl !! n
a009998_tabl = map reverse a009999_tabl
-- Reinhard Zumkeller, Feb 02 2014

E := (n,x) -> `if`(n=0,1,x*(1-x)*diff(E(n-1,x),x)+E(n-1,x)*(1+(n-1)*x));
G := (n,x) -> E(n,x)/(1-x)^(n+1);
A009998 := (n,k) -> coeff(series(G(n-k,x),x,18),x,k);
seq(print(seq(A009998(n,k),k=0..n)),n=0..6);
# Peter Luschny, Aug 02 2010

Flatten[Table[(j+1)^(i-j),{i,0,20},{j,0,i}]] (* Harvey P. Dale, Dec 25 2012 *)

T(i,j)=(j+1)^(i-j) \\ Charles R Greathouse IV, Feb 06 2017

T(n,n) = 1; T(n,k) = (k+1)*T(n-1,k) for k=0..n-1. - Reinhard Zumkeller, Feb 02 2014

T(n,m) = (m+1)*Sum_{k=0..n-m}((n+1)^(k-1)*(n-m)^(n-m-k)*(-1)^(n-m-k)*binomial(n-m-1,k-1)). - Vladimir Kruchinin, Sep 12 2015

a(62) corrected to 512 by T. D. Noe, Dec 20 2007

A055462 Superduperfactorials: product of first n superfactorials.

Original entry on oeis.org

1, 1, 2, 24, 6912, 238878720, 5944066965504000, 745453331864786829312000000, 3769447945987085350501386572267520000000000, 6916686207999802072984424331678589933649915805696000000000000000

Offset: 0

Henry Bottomley, Jun 26 2000

Next term has 92 digits and is too large to display.

Starting with offset 1, a(n) is a 'Matryoshka doll' sequence with alpha=1, the multiplicative counterpart to the additive A000332. The sequence for m with alpha<=m<=L is then computed as Prod_{n=alpha..m}(Prod_{k=alpha..n}(Prod_{i=alpha..k}(i))). - Peter Luschny, Jul 14 2009

			a(4) = 1!2!3!4!*1!2!3!*1!2!*1! = 288*12*2*1 = 6912.

Miles Seventh, Table of n, a(n) for n = 0..20
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.

Cf. A000142, A000178, A002109, A036740, A255268, A255269, A306635.

Cf. A057527, A057528, A260404.

[n eq 0 select 1 else (&*[j^Binomial(n-j+2,2): j in [1..n]]): n in [0..10]]; // G. C. Greubel, Jan 31 2024

seq(mul(mul(mul(i, i=alpha..k), k=alpha..n), n=alpha..m), m=alpha..10); # Peter Luschny, Jul 14 2009

Table[Product[BarnesG[j], {j, k + 1}], {k, 10}] (* Jan Mangaldan, Mar 21 2013 *)
Table[Round[Exp[(n+2)*(n+3)*(2*n+5)/8] * Exp[PolyGamma[-3, n+3]] * BarnesG[n+3]^(n+3/2) / (Glaisher^(n+3) * (2*Pi)^((n+3)^2/4) * Gamma[n+3]^((n+2)^2/2))], {n, 0, 10}] (* Vaclav Kotesovec, Feb 20 2015 after Jan Mangaldan *)
Nest[FoldList[Times,#]&,Range[0,15]!,2]  (* Harvey P. Dale, Jul 14 2023 *)

a(n)=my(t=1);prod(k=2,n,t*=k!) \\ Charles R Greathouse IV, Jul 28 2011

[product(j^binomial(n-j+2,2) for j in range(1,n+1)) for n in range(11)] # G. C. Greubel, Jan 31 2024

a(n) = a(n-1)*A000178(n) = Product_{i=1..n} (i!)^(n-i+1) = Product_{i=1..n} i^((n-i+1)*(n-i+2)/2).
log a(n) = (1/6) n^3 log n - (11/36) n^3 + O(n^2 log n). - Charles R Greathouse IV, Jan 13 2012
a(n) = exp((6 + 13 n + 9 n^2 + 2 n^3 - 8*(n + 2)*log(A)-2*(n + 2)^2*log(2*Pi) + 4*(2 n + 1)*logG(n + 2) - 4*(n + 1)^2*logGamma(n + 2) + 8*psi(-3, n + 2))/8) where A is the Glaisher-Kinkelin constant, logG(z) is the logarithm of the Barnes G function (A000178), and psi(-3, z) is a polygamma function of negative order (i.e., the second iterated integral of logGamma(z)). - Jan Mangaldan, Mar 21 2013
a(n) ~ exp(Zeta(3)/(8*Pi^2) - (2*n+3)*(11*n^2 + 24*n - 3)/72) * n^((2*n+3)*(2*n^2 + 6*n + 3)/24) * (2*Pi)^((n+1)*(n+2)/4) / A^(n+3/2), where A = A074962 = 1.28242712910062263687... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 20 2015

a(9) from N. J. A. Sloane, Dec 15 2008

cp's OEIS Frontend

A000312 a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Maxima

PARI

PARI

PARI

Python

Formula

A074962 Decimal expansion of Glaisher-Kinkelin constant A.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A000178 Superfactorials: product of first n factorials.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

PARI

PARI

Python

Python

Ruby

Formula

A002064 Cullen numbers: a(n) = n*2^n + 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A036740 a(n) = (n!)^n.

Views

Author

Keywords

Comments

References