cp's OEIS Frontend

A002109 Hyperfactorials: Product_{k = 1..n} k^k.

%I A002109 M3706 N1514 #193 Mar 19 2025 08:22:57
%S A002109 1,1,4,108,27648,86400000,4031078400000,3319766398771200000,
%T A002109 55696437941726556979200000,21577941222941856209168026828800000,
%U A002109 215779412229418562091680268288000000000000000,61564384586635053951550731889313964883968000000000000000
%N A002109 Hyperfactorials: Product_{k = 1..n} k^k.
%C A002109 A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 (this sequence) gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See refs Wikipedia and Szego, eq. (6.71.7). - _Alan Sokal_, Mar 02 2012
%C A002109 a(n) = (-1)^n/det(M_n) where M_n is the n X n matrix m(i,j) = (-1)^i/i^j. - _Benoit Cloitre_, May 28 2002
%C A002109 a(n) = determinant of the n X n matrix M(n) where m(i,j) = B(n,i,j) and B(n,i,x) denote the Bernstein polynomial: B(n,i,x) = binomial(n,i)*(1-x)^(n-i)*x^i. - _Benoit Cloitre_, Feb 02 2003
%C A002109 Partial products of A000312. - _Reinhard Zumkeller_, Jul 07 2012
%C A002109 Number of trailing zeros (A246839) increases every 5 terms since the exponent of the factor 5 increases every 5 terms and the exponent of the factor 2 increases every 2 terms. - _Chai Wah Wu_, Sep 03 2014
%C A002109 Also the number of minimum distinguishing labelings in the n-triangular honeycomb rook graph. - _Eric W. Weisstein_, Jul 14 2017
%C A002109 Also shows up in a term in the solution to the generalized version of Raabe's integral. - _Jibran Iqbal Shah_, Apr 24 2021
%D A002109 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
%D A002109 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.
%D A002109 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 477.
%D A002109 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002109 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002109 G. Szego, Orthogonal Polynomials, American Mathematical Society, 1981 edition, 432 Pages.
%H A002109 N. J. A. Sloane, <a href="/A002109/b002109.txt">Table of n, a(n) for n = 0..36</a>
%H A002109 Christian Aebi and Grant Cairns, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.433">Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials</a>, The American Mathematical Monthly 122.5 (2015): 433-443.
%H A002109 Mohammad K. Azarian, <a href="http://ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537.  Zbl 1133.11012.
%H A002109 blackpenredpen, <a href="https://www.youtube.com/watch?v=UDHGI-jRwUw">What is a Hyperfactorial?</a> Youtube video (2018).
%H A002109 CreativeMathProblems, <a href="https://www.youtube.com/watch?v=9xlVAusNrG4">A beautiful integral | Raabe's integral</a>, Youtube Video (2021).
%H A002109 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/glshkn/glshkn.html">Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for A002109, A000178) [Broken link]
%H A002109 Steven R. Finch, <a href="http://web.archive.org/web/20010622230958/http://www.mathsoft.com/asolve/constant/glshkn/glshkn.html">Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]
%H A002109 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_16">Fascinating Factorials</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
%H A002109 A. M. Ibrahim, <a href="http://www.nntdm.net/papers/nntdm-19/NNTDM-19-2-30_42.pdf">Extension of factorial concept to negative numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
%H A002109 Jeffrey C. Lagarias and Harsh Mehta, <a href="http://arxiv.org/abs/1409.4145">Products of binomial coefficients and unreduced Farey fractions</a>, arXiv:1409.4145 [math.NT], 2014.
%H A002109 Jean-Christophe Pain, <a href="https://arxiv.org/abs/2304.07629">Series representations for the logarithm of the Glaisher-Kinkelin constant</a>, arXiv:2304.07629 [math.NT], 2023.
%H A002109 Jean-Christophe Pain, <a href="https://arxiv.org/abs/2408.00353">Bounds on the p-adic valuation of the factorial, hyperfactorial and superfactorial</a>, arXiv:2408.00353 [math.NT], 2024. See p. 5.
%H A002109 Vignesh Raman, <a href="https://arxiv.org/abs/2012.00882">The Generalized Superfactorial, Hyperfactorial and Primorial functions</a>, arXiv:2012.00882 [math.NT], 2020.
%H A002109 Jonathan Sondow and Petros Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl., 332 (2007), 292-314; see Section 5.
%H A002109 László Tóth, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Toth/toth9.html">Weighted gcd-sum functions</a>, J. Integer Sequences, 14 (2011), Article 11.7.7.
%H A002109 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>.
%H A002109 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/K-Function.html">K-Function</a>.
%H A002109 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>.
%H A002109 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%H A002109 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F A002109 a(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
%F A002109 Determinant of n X n matrix m(i, j) = binomial(i*j, i). - _Benoit Cloitre_, Aug 27 2003
%F A002109 a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - _Peter Luschny_, Jun 23 2012
%F A002109 G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^k*x). - _Paul D. Hanna_, Oct 02 2013
%F A002109 a(n) = A240993(n) / A000142(n+1). - _Reinhard Zumkeller_, Aug 31 2014
%F A002109 a(n) ~ A * n^(n*(n+1)/2 + 1/12) / exp(n^2/4), where A = 1.2824271291006226368753425... is the Glaisher-Kinkelin constant (see A074962). - _Vaclav Kotesovec_, Feb 20 2015
%F A002109 a(n) = Product_{k=1..n} ff(n,k) where ff denotes the falling factorial. - _Peter Luschny_, Nov 29 2015
%F A002109 log a(n) = (1/2) n^2 log n - (1/4) n^2 + (1/2) n log n + (1/12) log n + log(A) + o(1), where log(A) = A225746 is the logarithm of Glaisher's constant. - _Charles R Greathouse IV_, Mar 27 2020
%F A002109 From _Amiram Eldar_, Apr 30 2023: (Start)
%F A002109 Sum_{n>=1} 1/a(n) = A347345.
%F A002109 Sum_{n>=1} (-1)^(n+1)/a(n) = A347352. (End)
%F A002109 From _Andrea Pinos_, Apr 04 2024: (Start)
%F A002109 a(n) = e^(Integral_{x=1..n+1} (x - 1/2 - log(sqrt(2*Pi)) + (n+1-x)*Psi(x)) dx), where Psi(x) is the digamma function.
%F A002109 a(n) = e^(Integral_{x=1..n} (x + 1/2 - log(sqrt(2*Pi)) + log(Gamma(x+1))) dx). (End)
%p A002109 f := proc(n) local k; mul(k^k,k=1..n); end;
%p A002109 A002109 := n -> exp(Zeta(1,-1,n+1)-Zeta(1,-1));
%p A002109 seq(simplify(A002109(n)),n=0..11); # _Peter Luschny_, Jun 23 2012
%t A002109 Table[Hyperfactorial[n], {n, 0, 11}] (* _Zerinvary Lajos_, Jul 10 2009 *)
%t A002109 Hyperfactorial[Range[0, 11]] (* _Eric W. Weisstein_, Jul 14 2017 *)
%t A002109 Join[{1},FoldList[Times,#^#&/@Range[15]]] (* _Harvey P. Dale_, Nov 02 2023 *)
%o A002109 (PARI) a(n)=prod(k=2,n,k^k) \\ _Charles R Greathouse IV_, Jan 12 2012
%o A002109 (PARI) a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1,k+1,(1+j^j*x+x*O(x^n)) )), n) \\ _Paul D. Hanna_, Oct 02 2013
%o A002109 (Haskell)
%o A002109 a002109 n = a002109_list !! n
%o A002109 a002109_list = scanl1 (*) a000312_list  -- _Reinhard Zumkeller_, Jul 07 2012
%o A002109 (Python)
%o A002109 A002109 = [1]
%o A002109 for n in range(1, 10):
%o A002109     A002109.append(A002109[-1]*n**n) # _Chai Wah Wu_, Sep 03 2014
%o A002109 (Sage)
%o A002109 a = lambda n: prod(falling_factorial(n,k) for k in (1..n))
%o A002109 [a(n) for n in (0..10)]  # _Peter Luschny_, Nov 29 2015
%Y A002109 Cf. A000178, A000142, A000312, A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374.
%Y A002109 Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].
%Y A002109 Cf. A001923, A051675, A240993, A255321, A255323, A255344.
%Y A002109 Cf. A246839 (trailing 0's).
%Y A002109 Cf. A347345, A347352.
%Y A002109 Cf. A261175 (number of digits).
%K A002109 nonn,easy,nice
%O A002109 0,3
%A A002109 _N. J. A. Sloane_