cp's OEIS Frontend

A000178 Superfactorials: product of first n factorials.

%I A000178 M2049 N0811 #329 Apr 22 2025 11:06:41
%S A000178 1,1,2,12,288,34560,24883200,125411328000,5056584744960000,
%T A000178 1834933472251084800000,6658606584104736522240000000,
%U A000178 265790267296391946810949632000000000,127313963299399416749559771247411200000000000,792786697595796795607377086400871488552960000000000000
%N A000178 Superfactorials: product of first n factorials.
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 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
%C A000178 All terms are in A064807 and all terms after a(2) are in A005101. - _Ivan N. Ianakiev_, Sep 02 2016
%C A000178 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
%C A000178 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
%C A000178 Also the automorphism group order of the n-triangular honeycomb rook graph. - _Eric W. Weisstein_, Jul 14 2017
%C A000178 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
%C A000178 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
%D A000178 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545.
%D A000178 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
%D A000178 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 A000178 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 231.
%D A000178 H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.
%D A000178 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000178 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000178 R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.
%H A000178 Boris Hostnik, <a href="/A000178/b000178.txt">Table of n, a(n) for n = 0..46</a>
%H A000178 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 A000178 Andreas B. G. Blobel, <a href="https://arxiv.org/abs/2203.09519">On convolution powers of 1/x</a>, arXiv:2203.09519 [math.CO], 2022.
%H A000178 E. F. Cornelius, Jr. and Phill Schultz, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Schultz/schultz14.html">Polynomial points </a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.
%H A000178 Selden Crary, <a href="http://arxiv.org/abs/1406.6326">Factorization of the Determinant of the Gaussian-Covariance Matrix of Evenly Spaced Points Using an Inter-dimensional Multiset Duality</a>, arXiv preprint arXiv:1406.6326 [math.ST], 2014-2019.
%H A000178 N. Destainville, R. Mosseri and F. Bailly, <a href="http://dx.doi.org/10.1007/BF02181243">Configurational Entropy of Codimension-One Tilings and Directed Membranes</a>, J. Stat. Phys. 87, Nos 3/4, 697 (1997).
%H A000178 J. East and R. D. Gray, <a href="http://arxiv.org/abs/1404.2359">Idempotent generators in finite partition monoids and related semigroups</a>, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
%H A000178 Richard Ehrenborg, <a href="http://www.jstor.org/stable/2589352">The Hankel determinant of exponential polynomials</a>, Amer. Math. Monthly, 107 (2000), 557-560.
%H A000178 William Q. Erickson and Jan Kretschmann, <a href="https://arxiv.org/abs/2311.07522">The structure and normalized volume of Monge polytopes</a>, arXiv:2311.07522 [math.CO], 2023. See p. 7.
%H A000178 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 A000178 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 A000178 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 A000178 Ivan Gutman, Wolfgang Linert, István Lukovits and Željko Tomović, <a href="http://dx.doi.org/10.1021/ci990060s">The multiplicative version of the Wiener index</a>, J. Chem. Inf. Comput. Sci., Vol. 40, No. 1 (2000), pp. 113-116.
%H A000178 Brady Haran and Sophie Maclean, <a href="https://www.youtube.com/watch?v=xV4A8oU3yew">What's special about 288?</a>, Numberphile video (2023).
%H A000178 Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H A000178 Nick Hobson, <a href="/A000178/a000178.py.txt">Python program for this sequence</a>.
%H A000178 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 A000178 Pavel L. Krapivsky, Jean-Marc Luck and Kirone Mallick, <a href="http://doi.org/10.1088/1742-5468/aaa79a">Quantum return probability of a system of N non-interacting lattice fermions</a>, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 2 (2018), 023104; <a href="https://arxiv.org/abs/1710.08178">arXiv preprint</a>, arXiv:1710.08178 [cond-mat.mes-hall], 2017-2018.
%H A000178 Jeffrey C. Lagarias and Harsh Mehta, <a href="https://doi.org/10.1142/S1793042116500044">Products of binomial coefficients and unreduced Farey fractions</a>, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; <a href="http://arxiv.org/abs/1409.4145">arXiv preprint</a>, arXiv:1409.4145 [math.NT], 2014-2015.
%H A000178 Mogens Esrom Larsen, <a href="http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/wronskian-harmony">Wronskian Harmony</a>, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
%H A000178 John W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000178 Rick Mabry and Laura McCormick, <a href="https://www.austms.org.au/wp-content/uploads/Gazette/2009/Nov09/TechPaperMabry.pdf">Square products of punctured sequences of factorials</a>, Gaz. Aust. Math. Soc., 2009.
%H A000178 Rémy Mosseri and Francis Bailly, <a href="http://dx.doi.org/10.1142/S0217979293002419">Configurational Entropy in Octagonal Tiling Models</a>, Int. J. Mod. Phys. B, Vol. 7, No. 6-7 (1993), pp. 1427-1436.
%H A000178 Rémy Mosseri, F. Bailly and C. Sire, <a href="http://dx.doi.org/10.1016/0022-3093(93)90342-U">Configurational Entropy in Random Tiling Models</a>, J. Non-Cryst. Solids, Vol. 153-154 (1993), pp. 201-204.
%H A000178 Amarnath Murthy, <a href="http://vixra.org/abs/1403.0675">Miscellaneous Results and Theorems on Smarandache terms and factor partitions</a>, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
%H A000178 Amarnath Murthy and Charles Ashbacher, <a href="http://www.gallup.unm.edu/~smarandache/MurthyBook.pdf">Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences</a>, Hexis, Phoenix; USA 2005. See Section 3.14.
%H A000178 Christian Radoux, <a href="https://www.ams.org/journals/notices/197804/197804FullIssue.pdf">Query 145</a>, Notices Amer. Math. Soc., 25-3 (1978), p. 197.
%H A000178 Christian Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Déterminants de Hankel et théorème de Sylvester</a>, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
%H A000178 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 A000178 Michel Waldschmidt, <a href="https://arxiv.org/abs/2504.14041">Schanuel Property for Elliptic and Quasi-Elliptic Functions</a>, arXiv:2504.14041 [math.NT], 2025. Mentions this sequence, see p. 10.
%H A000178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.
%H A000178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellNumber.html">Bell Number</a>.
%H A000178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FactorialProducts.html">Factorial Products</a>.
%H A000178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphAutomorphism.html">Graph Automorphism</a>.
%H A000178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasSequence.html">Lucas Sequence</a>.
%H A000178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Superfactorial.html">Superfactorial</a>.
%H A000178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VandermondeDeterminant.html">Vandermonde Determinant</a>.
%H A000178 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%H A000178 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%H A000178 <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.
%F A000178 a(0) = 1, a(n) = n!*a(n-1). - _Lee Hae-hwang_, May 13 2003, corrected by _Ilya Gutkovskiy_, Jul 30 2016
%F A000178 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]
%F A000178 a(n) = sqrt(A055209(n)). - _Philippe Deléham_, Mar 07 2004
%F A000178 a(n) = Product_{i=1..n} Product_{j=0..i-1} (i-j). - _Paul Barry_, Aug 02 2008
%F A000178 log a(n) = 0.5*n^2*log n - 0.75*n^2 + O(n*log n). - _Charles R Greathouse IV_, Jan 13 2012
%F A000178 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
%F A000178 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
%F A000178 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
%F A000178 G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k!*x). - _Paul D. Hanna_, Oct 02 2013
%F A000178 A203227(n+1)/a(n) -> e, as n -> oo. - _Daniel Suteu_, Jul 30 2016
%F A000178 From _Ilya Gutkovskiy_, Jul 30 2016: (Start)
%F A000178 a(n) = G(n+2), where G(n) is the Barnes G-function.
%F A000178 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).
%F A000178 Sum_{n>=0} (-1)^n/a(n) = A137986. (End)
%F A000178 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
%F A000178 Sum_{n>=0} 1/a(n) = A287013 = 1/A137987. - _Amiram Eldar_, Nov 19 2020
%F A000178 a(n) = Wronskian(1, x, x^2, ..., x^n). - _Mohammed Yaseen_, Aug 01 2023
%F A000178 From _Andrea Pinos_, Apr 04 2024: (Start)
%F A000178 a(n) = e^(Sum_{k=1..n} (Integral_{x=1..k+1} Psi(x) dx)).
%F A000178 a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + x*Psi(x)) dx).
%F A000178 a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + (n+1)*Psi(x) - log(Gamma(x))) dx).
%F A000178 Psi(x) is the digamma function. (End)
%e A000178 a(3) = (1/6)* | 1 1 1 | 2 4 8 | 3 9 27 |
%e A000178 a(7) = n! * a(n-1) = 7! * 24883200 = 125411328000.
%e A000178 a(12) = 1! * 2! * 3! * 4! * 5! * 6! * 7! * 8! * 9! * 10! * 11! * 12!
%e A000178 = 1^12 * 2^11 * 3^10 * 4^9 * 5^8 * 6^7 * 7^6 * 8^5 * 9^4 * 10^3 * 11^2 * 12^1
%e A000178 = 2^56 * 3^26 * 5^11 * 7^6 * 11^2.
%e A000178 G.f. = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 + ...
%p A000178 A000178 := proc(n)
%p A000178     mul(i!,i=1..n) ;
%p A000178 end proc:
%p A000178 seq(A000178(n),n=0..10) ; # _R. J. Mathar_, Oct 30 2015
%t A000178 a[0] := 1; a[1] := 1; a[n_] := n!*a[n - 1]; Table[a[n], {n, 1, 12}] (* _Stefan Steinerberger_, Mar 10 2006 *)
%t A000178 Table[BarnesG[n], {n, 2, 14}] (* _Zerinvary Lajos_, Jul 16 2009 *)
%t A000178 FoldList[Times,1,Range[20]!] (* _Harvey P. Dale_, Mar 25 2011 *)
%t A000178 RecurrenceTable[{a[n] == n! a[n - 1], a[0] == 1}, a, {n, 0, 12}] (* _Ray Chandler_, Jul 30 2015 *)
%t A000178 BarnesG[Range[2, 20]] (* _Eric W. Weisstein_, Jul 14 2017 *)
%o A000178 (PARI) A000178(n)=prod(k=2,n,k!) \\ _M. F. Hasler_, Sep 02 2007
%o A000178 (PARI) 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
%o A000178 (PARI) for(j=1,13, print1(prod(k=1,j,k^(j-k)),", ")) \\ _Hugo Pfoertner_, Apr 09 2020
%o A000178 (Maxima) A000178(n):=prod(k!,k,0,n)$ makelist(A000178(n),n,0,30); /* _Martin Ettl_, Oct 23 2012 */
%o A000178 (Ruby)
%o A000178 def mono_choices(a,b,n)
%o A000178     n - [a,b].max
%o A000178 end
%o A000178 def comm_mono_choices(n)
%o A000178     accum =1
%o A000178     0.upto(n-1) do |i|
%o A000178         i.upto(n-1) do |j|
%o A000178             accum = accum * mono_choices(i,j,n)
%o A000178         end
%o A000178     end
%o A000178     accum
%o A000178 end
%o A000178 1.upto(12) do |k|
%o A000178     puts comm_mono_choices(k)
%o A000178 end # _Chad Brewbaker_, Nov 03 2013
%o A000178 (Magma) [&*[Factorial(k): k in [0..n]]: n in [0..20]]; // _Bruno Berselli_, Mar 11 2015
%o A000178 (Python)
%o A000178 A000178_list, n, m = [1], 1,1
%o A000178 for i in range(1,100):
%o A000178     m *= i
%o A000178     n *= m
%o A000178     A000178_list.append(n) # _Chai Wah Wu_, Aug 21 2015
%o A000178 (Python)
%o A000178 from math import prod
%o A000178 def A000178(n): return prod(i**(n-i+1) for i in range(2,n+1)) # _Chai Wah Wu_, Nov 26 2023
%Y A000178 Partial products of A000142.
%Y A000178 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.
%Y A000178 A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
%Y A000178 A000178 is the Hankel transform (see A001906 for definition) of A000085, A000110, A000296, A005425, A005493, A005494 and A045379. - _John W. Layman_, Jul 28 2000
%Y A000178 Cf. A255322, A255358, A255359, A255360.
%Y A000178 Cf. A051675, A255321, A255323, A255344, A287013.
%Y A000178 Cf. A348692, A349079.
%K A000178 nonn,nice,easy
%O A000178 0,3
%A A000178 _N. J. A. Sloane_