0,3

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 (benoit7848c(AT)orange.fr), 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 (benoit7848c(AT)orange.fr), 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 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 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 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

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

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

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

Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: (Start)

Starting with offset 1 this is a 'Matryoshka doll' sequence with alpha=1, the mutiplicative counterpart to the additive A000292.

seq(mul(mul(i,i=alpha..k),k=alpha..n),n=alpha..12). (End)

E. F. Cornelius, Jr. and Phill Schultz, Polynomial points, Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.

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.

M. E. Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.

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).

Boris Hostnik, Table of n, a(n) for n=0...50

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, Link to a section of The World of Mathematics

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics

Eric Weisstein's World of Mathematics, Lucas Sequence

Eric Weisstein's World of Mathematics, Bell Number

Eric Weisstein's World of Mathematics, Factorial Products

Index entries for sequences related to factorial numbers

E. F. Cornelius, Jr. and Phill Schultz, Polynomial points , Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.

Eric Weisstein's World of Mathematics, Superfactorials.

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)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 07 2004

a(n)=product{i=1..n, product{j=0..i-1, i-j}}; [From Paul Barry (pbarry(AT)wit.ie), Aug 02 2008]

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.

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 (stefan.steinerberger(AT)gmail.com), 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

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

Table[BarnesG[n], {n, 2, 14}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2009]

(PARI) A000178(n)=prod(k=2, n, k!) \ - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Sep 02 2007

Cf. A002109, A000142, A036561.

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 (layman(AT)math.vt.edu), Jul 28 2000

Cf. A000292.

Cf. A098694, A098695.

Cf. A113271, A087316, A113208, A113231, A113257, A113258, A113320, A113336, A113498, A113173, A113170, A113475, A113492, A113497, A113533, A113534, A113535, A113153, A113154, A113122.

Cf. A114045.

Cf. A055462.

Sequence in context: A003121 A057170 A008338 this_sequence A108395 A009669 A012380

Adjacent sequences: A000175 A000176 A000177 this_sequence A000179 A000180 A000181

easy,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

One more term from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 10 2006