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%I M1953 N0773 #167 Aug 28 2023 08:28:28
%S 1,1,2,9,96,2500,162000,26471025,11014635520,11759522374656,
%T 32406091200000000,231627686043080250000,4311500661703860387840000,
%U 209706417310526095716965894400,26729809777664965932590782608648192
%N a(n) = Product_{k=1..n} k^(2k - 1 - n).
%C Absolute value of determinant of triangular matrix containing binomial coefficients.
%C These are also the products of consecutive horizontal rows of the Pascal triangle. - Jeremy Hehn (ROBO_HEN5000(AT)rose.net), Mar 29 2007
%C Limit_{n->oo} a(n)*a(n+2)/a(n+1)^2 = e, as follows from lim_{n->oo} (1 + 1/n)^n = e. - _Harlan J. Brothers_, Nov 26 2009
%C A000225 gives the positions of odd terms. - _Antti Karttunen_, Nov 02 2014
%C Product of all unreduced fractions h/k with 1 <= k <= h <= n. - _Jonathan Sondow_, Aug 06 2015
%C 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
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001142/b001142.txt">Table of n, a(n) for n = 0..50</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Mohammad K. Azarian, <a href="http://ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
%H Harlan J. Brothers, <a href="http://www.brotherstechnology.com/docs/Finding_e_in_Pascals_Triangle.pdf">Finding e in Pascal's Triangle</a>, Mathematics Magazine, 85 (2012), p. 51.
%H Harlan J. Brothers, <a href="http://dx.doi.org/10.1017/S0025557200004204">Pascal's Triangle: The Hidden Stor-e</a>, The Mathematical Gazette, 96 (2012), 145-148.
%H Jeffrey C. Lagarias and Harsh Mehta, <a href="http://arxiv.org/abs/1409.4145">Products of binomial coefficients and unreduced Farey fractions</a>, arXiv:1409.4145 [math.NT], 2014.
%H Leroy Quet, <a href="http://www.jstor.org/stable/2691040">Problem 1636</a>, Mathematics Magazine, Dec. 2001, p. 403.
%F a(n) = C(n, 0)*C(n, 1)* ... *C(n, n).
%F From _Harlan J. Brothers_, Nov 26 2009: (Start)
%F a(n) = Product_{j=1..n-2} Product_{k=1..j} (1+1/k)^k, n >= 3.
%F a(1) = a(2) = 1, a(n) = a(n-1) * Product_{k=1..n-2} (1+1/k)^k. (End)
%F a(n) = hyperfactorial(n)/superfactorial(n) = A002109(n)/A000178(n). - _Peter Luschny_, Jun 24 2012
%F 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
%F a(n) = Product_{i=1..n} Product_{j=1..i} (i/j). - _Pedro Caceres_, Apr 06 2019
%F a(n) = Product_{k=1..n} (n - k + 1)^(n - 2*k + 1). - _Harlan J. Brothers_, Aug 26 2023
%p a:=n->mul(binomial(n,k), k=0..n): seq(a(n), n=0..14); # _Zerinvary Lajos_, Jan 22 2008
%t Table[Product[k^(2*k - 1 - n), {k, n}], {n, 0, 20}] (* _Harlan J. Brothers_, Nov 26 2009 *)
%t Table[Hyperfactorial[n]/BarnesG[n+2], {n, 0, 20}] (* _Peter Luschny_, Nov 29 2015 *)
%t Table[Product[(n - k + 1)^(n - 2 k + 1), {k, 1, n}], {n, 0, 20}] (* _Harlan J. Brothers_, Aug 26 2023 *)
%o (PARI) for(n=0,16,print(prod(m=1,n,binomial(n,m))))
%o (PARI) A001142(n) = prod(k=1, n, k^((k+k)-1-n)); \\ _Antti Karttunen_, Nov 02 2014
%o (Scheme)
%o (define (A001142 n) (mul (lambda (k) (expt k (+ k k -1 (- n)))) 1 n))
%o (define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
%o ;; _Antti Karttunen_, Oct 28 2014
%o (Haskell)
%o a001142 = product . a007318_row -- _Reinhard Zumkeller_, Mar 16 2015
%o (Sage)
%o a = lambda n: prod(k^k/factorial(k) for k in (1..n))
%o [a(n) for n in range(20)] # _Peter Luschny_, Nov 29 2015
%o (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 */
%o (Magma) [(&*[Binomial(n,k): k in [0..n]]): n in [0..15]]; // _G. C. Greubel_, May 23 2019
%o (GAP) List([0..15], n-> Product([0..n], k-> Binomial(n,k) )) # _G. C. Greubel_, May 23 2019
%o (Python)
%o from math import factorial, prod
%o from fractions import Fraction
%o def A001142(n): return prod(Fraction((k+1)**k,factorial(k)) for k in range(1,n)) # _Chai Wah Wu_, Jul 15 2022
%Y Cf. A000178, A002109, A007318, A000225, A056077, A249421, A187059 (2-adic valuation), A249343, A249345, A249346, A249347, A249151.
%Y Cf. also A004788, A056606 (squarefree kernel), A256113.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, May 01 2000
%E Better description from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
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