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A000178
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Superfactorials: product of first n factorials.
(Formerly M2049 N0811)
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138
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1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000, 265790267296391946810949632000000000, 127313963299399416749559771247411200000000000
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OFFSET
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0,3
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COMMENTS
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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
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
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). - 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 Perez 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
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REFERENCES
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E. F. Cornelius, Jr. and Phill Schultz, Polynomial points, Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.
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).
R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107 (2000), 557-560.
S. 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.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.
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; http://repository.wit.ie/1693/1/AoifeThesis.pdf
M. E. Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
R. Mosseri and F. Bailly, Configurational Entropy in Octagonal Tiling Models, Int. J. Mod. Phys. B, Vol 7, 6-7, 1427 (1993).
R. Mosseri, F. Bailly and C. Sire, Configurational Entropy in Random Tiling Models, J. Non-Cryst. Solids, 153-154, 201 (1993).
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.
C. Radoux, Query 145, Notices Amer. Math. Soc., 25 (1978), 197.
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).
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LINKS
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Boris Hostnik, Table of n, a(n) for n = 0..46
E. F. Cornelius, Jr. and Phill Schultz, Polynomial points , Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.
S. R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178)
Nick Hobson, Python program for this sequence
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
C. Radoux, Determinants de Hankel et theoreme de Sylvester
Eric Weisstein's World of Mathematics, Superfactorial
Eric Weisstein's World of Mathematics, Barnes G-Function
Eric Weisstein's World of Mathematics, Vandermonde Determinant
Eric Weisstein's World of Mathematics, Lucas Sequence
Eric Weisstein's World of Mathematics, Bell Number
Eric Weisstein's World of Mathematics, Factorial Products Eric Weisstein's World of Mathematics, Superfactorials.
Index to divisibility sequences
Index entries for sequences related to factorial numbers
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FORMULA
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a(0) = 1, a(n+1) = n!*a(n). - Lee Hae-hwang (mathmaniac(AT)empal.com), May 13 2003
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
a(n) = sqrt(A055209(n)). - Philippe Deléham, Mar 07 2004
a(n) = product{i=1..n, product{j=0..i-1, i-j}}; [From 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) 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, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Oct 02 2012
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EXAMPLE
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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.
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MAPLE
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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
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, Mar 10 2006
seq(mul(mul(j, j=1..k), k=1..n), n=0..12); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 21 2007
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MATHEMATICA
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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}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2009]
FoldList[Times, 1, Range[20]!] (* From Harvey P. Dale, Mar 25 2011 *)
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PROG
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(PARI) A000178(n)=prod(k=2, n, k!) \ - M. F. Hasler, Sep 02 2007
(Maxima) A000178(n):=prod(k!, k, 0, n)$ makelist(A000178(n), n, 0, 30); /* Martin Ettl, Oct 23 2012 */
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CROSSREFS
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Cf. A002109, A000142, 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.
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
Sequence in context: A057170 A200564 A008338 * A108395 A009669 A202729
Adjacent sequences: A000175 A000176 A000177 * A000179 A000180 A000181
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KEYWORD
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easy,nonn,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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One more term from Stefan Steinerberger, Mar 10 2006
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STATUS
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approved
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