%I A000178 M2049 N0811
%S A000178 1,1,2,12,288,34560,24883200,125411328000,5056584744960000,
%T A000178 1834933472251084800000,6658606584104736522240000000,
%U A000178 265790267296391946810949632000000000,127313963299399416749559771247411200000000000
%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 by 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 (amarnath_murthy(AT)yahoo.com), 
               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 (benoit7848c(AT)orange.fr), 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 (benoit7848c(AT)orange.fr), 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 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               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 Lucus sequence U(i,Q), for any Q. When Q=0, the Vandermonde 
               matrix is obtained. - T. D. Noe (noe(AT)sspectra.com), 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). 
               - T. D. Noe (noe(AT)sspectra.com), 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 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               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 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: (Start)
%C A000178 Starting with offset 1 this is a 'Matryoshka doll' sequence with alpha=1, 
               the mutiplicative counterpart to the additive A000292.
%C A000178 seq(mul(mul(i,i=alpha..k),k=alpha..n),n=alpha..12). (End)
%D A000178 E. F. Cornelius, Jr. and Phill Schultz, Polynomial points, Journal of 
               Integer Sequences, Vol. 10 (2007), Article 07.3.6.
%D A000178 R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. 
               Math. Monthly, 107 (2000), 557-560.
%D A000178 S. 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 M. E. Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 
               1990, pp. 33-37.
%D A000178 Amarnath Murthy, Miscellaneous Results and Theorems on Smarandache terms 
               and factor partitions, Smarandache Notions Journal, Vol. 11, No. 
               1-2-3, Spring 2000.
%D A000178 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.
%D A000178 C. Radoux, Query 145, Notices Amer. Math. Soc., 25 (1978), 197.
%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).
%H A000178 Boris Hostnik, <a href="b000178.txt">Table of n, a(n) for n=0...50</a>
%H A000178 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/glshkn/glshkn.html">
               Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for 
               A002109, A000178)
%H A000178 Nick Hobson, <a href="a000178.py.txt">Python program for this sequence</
               a>
%H A000178 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%H A000178 C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">
               Determinants de Hankel et theoreme de Sylvester</a>
%H A000178 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Superfactorial.html">Link to a section of The World of Mathematics</
               a>
%H A000178 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BarnesG-Function.html">Link to a section of The World of Mathematics</
               a>
%H A000178 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               VandermondeDeterminant.html">Link to a section of The World of Mathematics</
               a>
%H A000178 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LucasSequence.html">Lucas Sequence</a>
%H A000178 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BellNumber.html">Bell Number</a>
%H A000178 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FactorialProducts.html">Factorial Products</a>
%H A000178 <a href="Sindx_Fa.html#factorial">Index entries for sequences related 
               to factorial numbers</a>
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Superfactorial.html">Superfactorials</a>.
%F A000178 a(0)=1, a(n+1)=n!*a(n). - Lee Hae-hwang (mathmaniac(AT)empal.com), May 
               13 2003
%F A000178 a(0) = 1, a(n) = 1^n*2^(n-1)*3^(n-2)...n = Prod {r^n-r+1}, r = 1 to n. 
               - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 12 2003
%F A000178 a(n) = sqrt(A055209(n)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 07 2004
%F A000178 a(n)=product{i=1..n, product{j=0..i-1, i-j}}; [From Paul Barry (pbarry(AT)wit.ie), 
               Aug 02 2008]
%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.
%p A000178 a[0]:=1:for n from 1 to 20 do a[n]:=product(k!, k=0..n) od: seq(a[n], 
               n=0..11); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 
               2007
%p A000178 a:=array(0...13): a[0]:=1: a[1]:=1:print(0,a[0]); print(1,a[1]); for 
               i from 2 to 13 do a[i]:= a[i-1]*(i!):print(i,a[i]); od: - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 27 2007 - Stefan Steinerberger 
               (stefan.steinerberger(AT)gmail.com), Mar 10 2006
%p A000178 seq(mul(mul(j,j=1..k), k=1..n), n=0..12); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Sep 21 2007
%t A000178 a[0] := 1; a[1] := 1; a[n_] := n!*a[n - 1]; Table[a[n], {n, 1, 12}] - 
               Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 10 
               2006 - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Mar 10 2006
%t A000178 Table[BarnesG[n], {n, 2, 14}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 16 2009]
%o A000178 (PARI) A000178(n)=prod(k=2,n,k!) \ - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), 
               Sep 02 2007
%Y A000178 Cf. A002109, A000142, A036561.
%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 (layman(AT)math.vt.edu), Jul 28 2000
%Y A000178 Cf. A000292.
%Y A000178 Cf. A098694, A098695.
%Y A000178 Cf. A113271, A087316, A113208, A113231, A113257, A113258, A113320, A113336, 
               A113498, A113173, A113170, A113475, A113492, A113497, A113533, A113534, 
               A113535, A113153, A113154, A113122.
%Y A000178 Cf. A114045.
%Y A000178 Cf. A055462.
%Y A000178 Sequence in context: A003121 A057170 A008338 this_sequence A108395 A009669 
               A012380
%Y A000178 Adjacent sequences: A000175 A000176 A000177 this_sequence A000179 A000180 
               A000181
%K A000178 easy,nonn,nice
%O A000178 0,3
%A A000178 N. J. A. Sloane (njas(AT)research.att.com).
%E A000178 One more term from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Mar 10 2006