A055209 -id:A055209 - cp's OEIS frontend

A059332 Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.

1, 1, 2, 24, 3456, 9953280, 859963392000, 3120635156889600000, 634153008009974906880000000, 9278496603801318870491332608000000000, 12218100099725239100847669366019325952000000000000, 1769792823810713244721831122736499011207487815680000000000000000

Offset: 0

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

nonn

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
Also called the bouncing factorial function. - Alexander Goebel, Apr 08 2020

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..32
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Googology Wiki, Bouncing Factorial

Cf. A010790, A385867.

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

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;
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)*n!^2/n)
    end:
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 29 2020

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

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

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

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

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

More terms from James Sellers, Jan 29 2001
Offset corrected. Comment and formula aligned with new offset by Johannes W. Meijer, Jun 24 2009
a(0)=1 prepended by Alois P. Heinz, Apr 08 2020

A162014 Sequence related to the o.g.f.s. of the right hand columns of the EG1 triangle A162005.

Original entry on oeis.org

1, 8, -1536, -14155776, 10436770529280, 923378661099307008000, -13724698564186788948502118400000, -45695540009113634492156662349750599680000000

Offset: 1

Johannes W. Meijer, Jun 27 2009

The a(n) are the sums of the coefficients of the polynomials that appear in the numerators of the o.g.f.s. of the right hand columns of the EG1 triangle A162005, see the examples.

			The polynomials in the numerators of the first few o.g.f.s are:
numer(GF(1)) = 1
numer(GF(2)) = 2+6*z
numer(GF(3)) = 16+296*z-768*z^2-1080*z^3
numer(GF(4)) = 272+17376*z-321360*z^2-1298624*z^3+8914800*z^4-11262240*z^5-10206000*z^6
numer(GF(5)) = 7936 + 1305088*z - 79792256*z^2 - 109331968*z^3 + 41828672000*z^4-460917924352*z^5 + 238697445120*z^6 + 5066784271872*z^7 - 14723693948160*z^8+ 12172737024000*z^9 + 8101522800000*z^10

A000012, A004004 (2x), A162008, A162009 and A162010 are the first five right hand columns of the EG1 triangle A162005.

Cf. A055209 and A059332.

a(n) = (-1)^( (n^2+n-2)/2)*4^((n-1)*n/2)*n!*product(k!, k=0..n-1)^2

A239350 Superprimorials squared.

Original entry on oeis.org

1, 4, 144, 129600, 5715360000, 30497732496000000, 27502882612852046400000000, 7167813920637790505994548640000000000, 674376505248717910810215697948155164304000000000000, 33564007734235791949707248640534383334045138980782017600000000000000

Offset: 0

Jonathan Sondow, Mar 22 2014

nonn

Square of product of first n primorials = A006939(n)^2.
Smallest number with n distinct even exponents in its prime factorization.
The prime version of Ramanujan's infinite nested radical 1*sqrt(1+2*sqrt(1+3*sqrt(1+...))) is 2*sqrt(1+3*sqrt(1+5*sqrt(1+...))) = sqrt(4+sqrt(144+sqrt(129600+...))) = sqrt(a(1)+sqrt(a(2)+sqrt(a(3)+...))). See A239349 and A055209.

Vincenzo Librandi, Table of n, a(n) for n = 0..27

Cf. A002110, A055209, A006939, A239349.

Rest[FoldList[Times, 1, FoldList[Times, 1, Prime[Range[9]]^2]]]

a(n) = Product_{k=1..n} A002110(k)^2 = Product_{k=1..n} prime(k)^(2(n-k+1)).

A100685 Powers of factorials A000142.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 24, 32, 36, 64, 120, 128, 216, 256, 512, 576, 720, 1024, 1296, 2048, 4096, 5040, 7776, 8192, 13824, 14400, 16384, 32768, 40320, 46656, 65536, 131072, 262144, 279936, 331776, 362880, 518400, 524288, 1048576, 1679616, 1728000

Offset: 1

Kyle Schalm and Jonathan Sondow, Dec 08 2004

Subsequence of A001013. Supersequence of A036740 without its first term.

Supersequence also of A046882 and A055209 without their first terms. - Jonathan Sondow and Robert G. Wilson v, Dec 19 2004

			24 = (4!)^1 and 36 = (3!)^2.

G. C. Greubel, Table of n, a(n) for n = 1..9660
Index entries for sequences related to factorial numbers.

Cf. A000142, A001013, A036740, A331373.
Cf. also A046882 and A055209.
Subsequences: A000079, A000400, A009968.

With[{ln = Log[10!]}, Table[With[{f = m!}, Table[f^j, {j, 0, Floor[ln/Log[f]]}]], {m, 2, 10}]] //Flatten //Union

Sum_{n>=1} 1/a(n) = 1 + A331373. - Amiram Eldar, Nov 21 2021

A101800 a(n)= abs(det[A000166(i+j+1)]), i,j=0...n, is the absolute value of the Hankel determinant of order n+1 of the derangements numbers, cf. A000166.

Original entry on oeis.org

0, 1, 16, 2160, 4644864, 220962816000, 126311423016960000, 97655159393202733056000000, 2873961139404949958783139840000000000, 5118723340142578530942677236206891171840000000000

Offset: 0

Karol A. Penson, Dec 17 2004

nonn

a(n) = abs(product( (p!)^2,p=0..n )*(n+1)!*LaguerreL(n+1,0,1)), n=0,1..., where LaguerreL(n,lambda,x) are generalized Laguerre polynomial.

Cf. A000166, A055209, A009940, A101799.

a[n_] := Table[Subfactorial[i+j+1], {i, 0, n}, {j, 0, n}] // Det // Abs;
Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Aug 18 2024 *)

a(n) = abs(A055209(n)*A009940(n+1)). [corrected by Vaclav Kotesovec, Feb 25 2019]

A136411 a(n) = Product_{k=1..n} (2k-1)^(2n-2*k+1).

Original entry on oeis.org

1, 3, 135, 212625, 21097715625, 207248662456171875, 291128066470548703880859375, 79746389028864195813528714933837890625, 5570294521107277357810167397301815834831695556640625

Offset: 1

Ctibor O. Zizka, Mar 31 2008

G. C. Greubel, Table of n, a(n) for n = 1..29

Cf. A136807, A087315, A000178, A055209.

[(&*[(2*k-1)^(2*n-2*k+1): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[(2*k - 1)^(2*n - 2*k + 1), {k, 1, n}], {n, 1, 10}] (* Stefan Steinerberger, May 18 2008 *)
sf[n_] := BarnesG[n + 2]; a[n_] := sf[2 n - 1]/(2^(n (n - 1)) sf[n - 1]^2); Table[a[n], {n, 1, 10}]  (* Robert Coquereaux, Apr 02 2013 *)

a(n) = prod(k=1, n, (2*k-1)^(2*n-2*k+1)) \\ Anders Hellström, Sep 16 2015

a(n) = A107254(n) / 2^(n*(n - 1)).
a(n) = sf(2*n-1) / (2^(n*(n-1)) * sf(n-1)^2), n >= 1, where sf(n) = BarnesG(n + 2) is the superfactorial defined in A000178. - Robert Coquereaux, Apr 02 2013
a(n) ~ A * 2^(n^2 + n - 1/12) * n^(n^2 + 1/12) / exp(3*n^2/2 + 1/12), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

More terms from Stefan Steinerberger, May 18 2008

A136807 Hankel transform of double factorial numbers n!*2^n=A000165(n).

Original entry on oeis.org

1, 4, 256, 589824, 86973087744, 1282470362637926400, 2723154477021188283432960000, 1133321924829207204666583887642624000000, 120746421332702772771144114237731253721340313600000000

Offset: 0

Paul Barry, Jan 23 2008

By the properties of the Hankel transform, a(n)=2^(n(n+1))*A055209(n).

Also Hankel transform of A000354, A010844, A082032. - Philippe Deléham, Jan 23 2008

G. C. Greubel, Table of n, a(n) for n = 0..28
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000354, A010844, A082032, A055209.

[1] cat [(&*[(2*k)^(2*(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[(2k)^(2(n-k+1)),{k,n}],{n,0,10}] (* Harvey P. Dale, Apr 11 2013 *)

for(n=0,10, print1(prod(k=1,n,(2*k)^(2*(n-k+1))), ", ")) \\ G. C. Greubel, Oct 14 2018

a(n) = Product_{k=1..n} (2k)^(2(n-k+1)).

a(n) ~ 2^((n+1)^2) * Pi^(n+1) * n^(n^2 + 2*n + 5/6) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Feb 24 2019

A290770 a(n) = Product_{k=1..n} k^(2*k).

Original entry on oeis.org

1, 1, 16, 11664, 764411904, 7464960000000000, 16249593066946560000000000, 11020848942410302096869949440000000000, 3102093199396597590886754340698424229232640000000000, 465607547420733489126893933985879279492195953053596584509440000000000

Offset: 0

Ilya Gutkovskiy, Aug 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..27
Eric Weisstein's World of Mathematics, Hyperfactorial
Index entries for sequences related to factorial numbers

Cf. A000178, A001044, A001818, A002109, A051675, A055209, A061742, A062206, A074962, A184877, A185141, A260122.

[1] cat [(&*[k^(2*k): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[k^(2 k), {k, 1, n}], {n, 0, 9}]
Table[Hyperfactorial[n]^2, {n, 0, 9}]
Table[n!^(2 n)/BarnesG[n + 1]^2, {n, 0, 9}]

a(n) = prod(k=1, n, k^(2*k)) \\ Felix Fröhlich, Aug 10 2017

a(n) = A002109(n)^2.
a(n) = A185141(n)/A000178(n-1)^2 for n > 0.
a(n) = (n!)^(2*n)/G(n+1)^2, where G() is the Barnes G-function.
a(n) ~ A^2*exp(-n^2/2)*n^(n*(n+1))*n^(1/6), where A is the Glaisher-Kinkelin constant (A074962).

A067736 Decimal expansion of exp(3/2).

Original entry on oeis.org

4, 4, 8, 1, 6, 8, 9, 0, 7, 0, 3, 3, 8, 0, 6, 4, 8, 2, 2, 6, 0, 2, 0, 5, 5, 4, 6, 0, 1, 1, 9, 2, 7, 5, 8, 1, 9, 0, 0, 5, 7, 4, 9, 8, 6, 8, 3, 6, 9, 6, 6, 7, 0, 5, 6, 7, 7, 2, 6, 5, 0, 0, 8, 2, 7, 8, 5, 9, 3, 6, 6, 7, 4, 4, 6, 6, 7, 1, 3, 7, 7, 2, 9, 8, 1, 0, 5, 3, 8, 3, 1, 3, 8, 2, 4, 5, 3, 3, 9, 1, 3, 8, 8, 6, 1

Offset: 1

Alford Arnold, Mar 10 2002

It is well known that derangements, A000166, are related to exp(1) (cf. A001113). It appears that derangements with minimal cycle size 3 relate to exp(1+1/2). for example, 720/160 = 4.5, 5040/1140 = 4.4210, 40320/8988 = 4.4859, 362880/80864 = 4.4875 the pattern continues - derangements with minimal cycle size 4 appear to relate in the same way to exp(1 + 1/2 +1/3).

			4.4816890703380648226020554601192758190057498683696...

D. M. Bătinetu-Giurgiu, Problem 4179, Crux Mathematicorum, Vol. 42, No. 8 (2016), p. 357; Solution to Problem 4179 by Kee-Wai Lau, ibid., Vol. 43, No. 8 (2017), p. 369.
Index entries for transcendental numbers

Cf. A000142, A000166, A001113, A038205, A047865, A055209.

RealDigits[Exp[3/2],10,120][[1]] (* Harvey P. Dale, Apr 24 2016 *)

exp(3/2) \\ Charles R Greathouse IV, Nov 21 2024

Equals lim_{n->oo} n/A055209(n)^(1/n^2) (Bătinetu-Giurgiu, 2016). - Amiram Eldar, Apr 11 2022

Solution of x = Integral_{t=0..x} log(t^2) dt. - Thomas Scheuerle, Sep 22 2023

More terms from Sascha Kurz, Mar 19 2002

A091810 Hankel transform of the sequence A001469 (unsigned), Genocchi numbers of first kind.

Original entry on oeis.org

1, 2, 96, 497664, 825564856320, 1027134771639091200000, 1932215036193527461576704000000000, 9973959265081827837426668870219857920000000000000

Offset: 0

Philippe Deléham, Mar 07 2004

nonn

This sequence is the Hankel transform (see A001906 for definition)of the sequence defined by, for n>=0, a(n) = |A001469(n+1)|; example: det([1, 1, 3, 17; 1, 3, 17, 155; 3, 17, 155, 2073; 17, 155, 2073, 38227]) = 497664 = 4!*12^4.

G. C. Greubel, Table of n, a(n) for n = 0..24

Cf. A000178 A001469 A001906 A055209.

[Factorial(n+1)*(&*[(Factorial(k))^2: k in [0..n]])^2: n in [0..10]]; // G. C. Greubel, Oct 14 2018

Table[(n + 1)!*BarnesG[n + 2]^4, {n, 0, 10}] (* G. C. Greubel, Oct 14 2018 *)

for(n=0,10, print1((n+1)!*(prod(k=0,n, (k!)^2))^2, ", ")) \\ G. C. Greubel, Oct 14 2018

a(n) = (n+1)!*A000178(n)^4 = (n+1)!*A055209(n)^2.

cp's OEIS Frontend

A059332 Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.

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A162014 Sequence related to the o.g.f.s. of the right hand columns of the EG1 triangle A162005.

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A239350 Superprimorials squared.

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Mathematica

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A100685 Powers of factorials A000142.

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A101800 a(n)= abs(det[A000166(i+j+1)]), i,j=0...n, is the absolute value of the Hankel determinant of order n+1 of the derangements numbers, cf. A000166.

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A136411 a(n) = Product_{k=1..n} (2*k-1)^(2*n-2*k+1).

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Magma

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A136807 Hankel transform of double factorial numbers n!*2^n=A000165(n).

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A290770 a(n) = Product_{k=1..n} k^(2*k).

A136411 a(n) = Product_{k=1..n} (2k-1)^(2n-2*k+1).