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A001142
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a(n) = Product_{k=1..n} k^(2k - 1 - n).
(Formerly M1953 N0773)
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58
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1, 1, 2, 9, 96, 2500, 162000, 26471025, 11014635520, 11759522374656, 32406091200000000, 231627686043080250000, 4311500661703860387840000, 209706417310526095716965894400, 26729809777664965932590782608648192
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OFFSET
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0,3
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COMMENTS
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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
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
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
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REFERENCES
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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).
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LINKS
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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].
Leroy Quet, Problem 1636, Mathematics Magazine, Dec. 2001, p. 403.
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FORMULA
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a(n) = C(n, 0)*C(n, 1)* ... *C(n, n).
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) ~ 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
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MAPLE
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a:=n->mul(binomial(n, k), k=0..n): seq(a(n), n=0..14); # Zerinvary Lajos, Jan 22 2008
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MATHEMATICA
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Table[Hyperfactorial[n]/BarnesG[n+2], {n, 0, 20}] (* Peter Luschny, Nov 29 2015 *)
Table[Product[(n - k + 1)^(n - 2 k + 1), {k, 1, n}], {n, 0, 20}] (* Harlan J. Brothers, Aug 26 2023 *)
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PROG
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(PARI) for(n=0, 16, print(prod(m=1, n, binomial(n, m))))
(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)))))))
(Haskell)
(Sage)
a = lambda n: prod(k^k/factorial(k) for k in (1..n))
(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
(Python)
from math import factorial, prod
from fractions import Fraction
def A001142(n): return prod(Fraction((k+1)**k, factorial(k)) for k in range(1, n)) # Chai Wah Wu, Jul 15 2022
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CROSSREFS
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Cf. A000178, A002109, A007318, A000225, A056077, A249421, A187059 (2-adic valuation), A249343, A249345, A249346, A249347, A249151.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Better description from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
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STATUS
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approved
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