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A000178 Superfactorials: product of first n factorials.
(Formerly M2049 N0811)
145
1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000, 265790267296391946810949632000000000, 127313963299399416749559771247411200000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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). - 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

a(n-1) = product(j!,j=1..n-1) = V(n) = product(j-i,1<= i < j <= n) (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

REFERENCES

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

A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42; http://www.nntdm.net/papers/nntdm-19/NNTDM-19-2-30_42.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).

R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.

LINKS

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.

J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359, 2014

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

FORMULA

a(0) = 1, a(n+1) = n!*a(n). - Lee Hae-hwang, 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, 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) 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

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

EXAMPLE

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.

MAPLE

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, 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, Mar 27 2007 - Stefan Steinerberger, Mar 10 2006

seq(mul(mul(j, j=1..k), k=1..n), n=0..12); - Zerinvary Lajos, Sep 21 2007

MATHEMATICA

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 *)

LinearRecurrence[{#1!}, {1}, 13] (* Robert G. Wilson v, Jun 15 2013 *)

PROG

(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 */

(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

(Ruby)

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

CROSSREFS

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

KEYWORD

easy,nonn,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

One more term from Stefan Steinerberger, Mar 10 2006

STATUS

approved

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Last modified July 30 19:06 EDT 2014. Contains 245074 sequences.