login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001142 a(n) = Product_{k=1..n} k^(2k - 1 - n).
(Formerly M1953 N0773)
51
1, 1, 2, 9, 96, 2500, 162000, 26471025, 11014635520, 11759522374656, 32406091200000000, 231627686043080250000, 4311500661703860387840000, 209706417310526095716965894400, 26729809777664965932590782608648192 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Absolute value of determinant of triangular matrix containing binomial coefficients.

These are also the products of consecutive horizontal rows of the Pascal triangle. - Jeremy Hehn (ROBO_HEN5000(AT)rose.net), Mar 29 2007

Lim_{n->infinity} a(n)*a(n+2)/a(n+1)^2 = e, as follows from lim_{n->infinity} (1 + 1/n)^n = e. - Harlan J. Brothers, Nov 26 2009

A000225 gives the positions of odd terms. - Antti Karttunen, Nov 02 2014

Product of all unreduced fractions h/k with 1 <= k <= h <= n. - Jonathan Sondow, Aug 06 2015

a(n) is a product of the binomial coefficients from the n-th row of the Pascal triangle, for n= 0, 1, 2, ... For n > 0, a(n) means the number of all maximum chains in the poset formed by the n-dimensional Boolean cube {0,1}^n and the relation "precedes by weight". This relation is defined over {0,1}^n as follows: for arbitrary vectors u, v of {0,1}^n we say that "u precedes by weight v" if wt(u) < wt(v) or if u = v, where wt(u) denotes the (Hamming) weight of u. For more details, see the sequence A051459. - Valentin Bakoev, May 17 2019

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..50

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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.

H. J. Brothers, Finding e in Pascal's Triangle, Mathematics Magazine, 85 (2012), p. 51.

H. J. Brothers, Pascal's Triangle: The Hidden Stor-e, The Mathematical Gazette, 96 (2012), 145-148.

Jeffrey C. Lagarias, Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Leroy Quet, Problem 1636, Mathematics Magazine, Dec. 2001, p. 403.

FORMULA

a(n) = C(n, 0)*C(n, 1)* ... *C(n, n).

From Harlan J. Brothers, Nov 26 2009: (Start)

a(n) = Product_{j=1..n-2} Product_{k=1..j} (1+1/k)^k, n >= 3.

a(1) = a(2) = 1, a(n) = a(n-1) * Product_{k=1..n-2} (1+1/k)^k. (End)

a(n) = hyperfactorial(n)/superfactorial(n) =  A002109(n)/A000178(n). - Peter Luschny, Jun 24 2012

a(n) ~ A^2 * exp(n^2/2 + n - 1/12) / (n^(n/2 + 1/3) * (2*Pi)^((n+1)/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

a(n) = Product_{i=1..n} Product_{j=1..i} (i/j). - Pedro Caceres, Apr 06 2019

MAPLE

a:=n->mul(binomial(n, k), k=0..n): seq(a(n), n=0..14); # Zerinvary Lajos, Jan 22 2008

MATHEMATICA

Table[Product[k^(2*k - 1 - n), {k, n}], {n, 0, 9}] (* Harlan J. Brothers, Nov 26 2009 *)

Table[Hyperfactorial[n]/BarnesG[n+2], {n, 0, 20}] (* Peter Luschny, Nov 29 2015 *)

PROG

(PARI) for(n=0, 16, print(prod(m=1, n, binomial(n, m))))

(PARI) A001142(n) = prod(k=1, n, k^((k+k)-1-n)); \\ Antti Karttunen, Nov 02 2014

(Scheme)

(define (A001142 n) (mul (lambda (k) (expt k (+ k k -1 (- n)))) 1 n))

(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))

;; Antti Karttunen, Oct 28 2014

(Haskell)

a001142 = product . a007318_row -- Reinhard Zumkeller, Mar 16 2015

(Sage)

a = lambda n: prod(k^k/factorial(k) for k in (1..n))

print [a(n) for n in range(20)] # Peter Luschny, Nov 29 2015

(Maxima) a(n):= prod(binomial(n, k), k, 0, n); n : 15; for i from 0 thru n do print (a(i)); /* Valentin Bakoev, May 17 2019 */

(MAGMA) [(&*[Binomial(n, k): k in [0..n]]): n in [0..15]]; // G. C. Greubel, May 23 2019

(GAP) List([0..15], n-> Product([0..n], k-> Binomial(n, k) )) # G. C. Greubel, May 23 2019

CROSSREFS

Cf. A000178, A002109, A007318, A000225, A056077, A249421, A187059 (2-adic valuation), A249343, A249345, A249346, A249347, A249151.

Cf. also A004788, A056606 (squarefree kernel), A256113.

Sequence in context: A106343 A086992 A115965 * A111847 A013132 A317275

Adjacent sequences:  A001139 A001140 A001141 * A001143 A001144 A001145

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, May 01 2000

Better description from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 14 15:10 EST 2019. Contains 329126 sequences. (Running on oeis4.)