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A002109 Hyperfactorials: Product_{k = 1..n} k^k.
(Formerly M3706 N1514)
28
1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, 21577941222941856209168026828800000, 215779412229418562091680268288000000000000000, 61564384586635053951550731889313964883968000000000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See refs Wikipedia and Szego, eq. (6.71.7). - Alan Sokal, Mar 02 2012

a(n) = (-1)^n/det(M_n) where M_n is the n X n matrix m(i,j) = (-1)^i/i^j - Benoit Cloitre, May 28 2002

a(n) = determinant of the n X n matrix M(n) where m(i,j) = B(n,i,j) and B(n,i,x) denote the Bernstein polynomial : B(n,i,x) = binomial(n,i)*(1-x)^(n-i)*x^i. - Benoit Cloitre, Feb 02 2003

The least significant non-zero digit of a(n): 1, 1, 4, 8, 8, 4, 4, 2, 2, 8, 8, 8, 8, 4, 4, 6, 6, 2, 8, 2, 2, 2, 8, 6, 6, 4, 4, 2, 2, 8, ... - Robert G. Wilson v, May 11 2012.

Partial products of A000312. - Reinhard Zumkeller, Jul 07 2012

a(n) = A240993(n) / A000142(n+1). - Reinhard Zumkeller, Aug 31 2014

Number of trailing zeros increases every 5 terms since the exponent of the factor 5 increases every 5 terms and the exponent of the factor 2 increases every 2 terms. - Chai Wah Wu, Sep 03 2014

REFERENCES

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. 477.

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).

G. Szego, Orthogonal Polynomials, American Mathematical Society, 1981 edition, 432 Pages.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..36

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537.  Zbl 1133.11012.

S. R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [At present this link does not work]

A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see Section 5.

Laszlo Toth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7

Eric Weisstein's World of Mathematics, Hyperfactorial

Eric Weisstein's World of Mathematics, K-Function

Wikipedia, Hermite polynomials

Index to divisibility sequences

Index entries for sequences related to factorial numbers

FORMULA

Determinant of n X n matrix m(i, j) = binomial(i*j, i). - Benoit Cloitre, Aug 27 2003

log a(n) = 0.5 n^2 log n + n^2/4 + O(n log n). [Charles R Greathouse IV, Jan 12 2012]

a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - Peter Luschny, Jun 23 2012

G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^k*x). - Paul D. Hanna, Oct 02 2013

MAPLE

f := proc(n) local k; mul(k^k, k=1..n); end;

a[0]:=1:for n from 1 to 20 do a[n]:=product(n!/k!, k=0..n) od: seq(a[n], n=0..11); # Zerinvary Lajos, Jun 11 2007

seq(mul(mul(k, j=1..k), k=1..n), n=0..11); # Zerinvary Lajos, Sep 21 2007

A002109 := n -> exp(Zeta(1, -1, n+1)-Zeta(1, -1)); seq(simplify(A002109(n)), n=0..11); # Peter Luschny, Jun 23 2012

MATHEMATICA

lst={}; s=1; Do[AppendTo[lst, s*=n^n], {n, 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)

Table[Hyperfactorial[n], {n, 0, 11}] (* Zerinvary Lajos, Jul 10 2009 *)

PROG

(PARI) a(n)=prod(k=2, n, k^k) \\ Charles R Greathouse IV, Jan 12 2012

(Haskell)

a002109 n = a002109_list !! n

a002109_list = scanl1 (*) a000312_list  -- Reinhard Zumkeller, Jul 07 2012

(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, (1+j^j*x+x*O(x^n)) )), n)} \\ Paul D. Hanna, Oct 02 2013

(Python)

A002109 = [1]

for n in range(1, 10):

....A002109.append(A002109[-1]*n**n) # Chai Wah Wu, Sep 03 2014

CROSSREFS

Cf. A000178, A000142.

A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.

Cf. A000312, A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374.

Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].

Cf. A240993.

Sequence in context: A107048 A185702 A212803 * A076265 A114876 A037980

Adjacent sequences:  A002106 A002107 A002108 * A002110 A002111 A002112

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 31 06:33 EDT 2014. Contains 248845 sequences.