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A001142 a(n) = product{k=1..n} k^(2k-1-n).
(Formerly M1953 N0773)
19
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->inf (a(n)a(n+2))/a(n+1)^2 = e, as follows from lim n->inf (1 +  1/n)^n = e. [Harlan J. Brothers, Nov 26 2009]

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.

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

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.

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[Product[(1+1/k)^k, {k, 1, j}], {j, 1, n-2}], n >= 3

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

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

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 *)

PROG

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

CROSSREFS

Sequence in context: A106343 A086992 A115965 * A111847 A013132 A013057

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

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Last modified October 22 11:41 EDT 2014. Contains 248397 sequences.