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A059332 Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n. 8
1, 2, 24, 3456, 9953280, 859963392000, 3120635156889600000, 634153008009974906880000000, 9278496603801318870491332608000000000, 12218100099725239100847669366019325952000000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Hankel transform of n! (A000142(n)) and of A003319. - Paul Barry, Oct 07 2008

Hankel transform of A000255. - Paul Barry, Apr 22 2009

Monotonic magmas of size n, i.e., magmas with elements labeled 1..n where product(i,j) >= max(i,j). - Chad Brewbaker, Nov 03 2013

REFERENCES

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

LINKS

Table of n, a(n) for n=1..10.

FORMULA

a(n) = a(n-1)*(n!)*(n-1)! for n >= 2 so a(n) = product k=1, 2, ..., n k!*(k-1)!.

a(n) = 2^C(n,2)*product{k=1..(n-1), C(k+2,2)^(n-1-k)}. - Paul Barry, Jan 15 2009

a(n) = n!*product(k!, k=0..n-1)^2. - Johannes W. Meijer, Jun 27 2009

a(n) ~ (2*Pi)^(n+1/2) * exp(1/6 - n - 3*n^2/2) * n^(n^2 + n + 1/3) / A^2, where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 01 2015

EXAMPLE

a(4) = 3456 because the relevant matrix is {1 2 6 24 / 2 6 24 120 / 6 24 120 720 / 24 120 720 5040 } and the determinant is 3456.

MAPLE

with(linalg): Digits := 500: A059332 := proc(n) local A, i, j: A := array(1..n, 1..n): for i from 1 to n do for j from 1 to n do A[i, j] := (i+j-1)! od: od: RETURN(det(A)) end: for n from 1 to 20 do printf(`%d, `, A059332(n)) od;

MATHEMATICA

Table[n! BarnesG[n+1]^2, {n, 1, 10}] (* Jean-Fran├žois Alcover, Sep 19 2016 *)

PROG

(Ruby)

def mono_choices(a, b, n)

    n - [a, b].max

end

def all_mono_choices(n)

    accum =1

    0.upto(n-1) do |i|

        0.upto(n-1) do |j|

            accum = accum * mono_choices(i, j, n)

        end

    end

    accum

end

1.upto(12) do |k|

puts all_mono_choices(k)

end # Chad Brewbaker, Nov 03 2013

(PARI) A059332(n)=matdet(matrix(n, n, i, j, (i+j-1)!)) \\ - M. F. Hasler, Nov 03 2013

(PARI) a(n) = 2^binomial(n, 2)*prod(k=1, n-1, binomial(k+2, 2)^(n-1-k)) \\ Ralf Stephan, Nov 04 2013

CROSSREFS

Cf. A010790.

Cf. A162014 and A055209. - Johannes W. Meijer, Jun 27 2009

Sequence in context: A111428 A111429 A111430 * A000794 A159907 A242484

Adjacent sequences:  A059329 A059330 A059331 * A059333 A059334 A059335

KEYWORD

nonn

AUTHOR

Noam Katz (noamkj(AT)hotmail.com), Jan 26 2001

EXTENSIONS

More terms from James A. Sellers, Jan 29 2001

Offset corrected. Comment and formula aligned with new offset by Johannes W. Meijer, Jun 24 2009

STATUS

approved

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Last modified February 21 02:05 EST 2018. Contains 299388 sequences. (Running on oeis4.)