A162014 -id:A162014 - cp's OEIS frontend

A004004 a(n) = (3^(2n+1) - 8n - 3)/16.

0, 1, 14, 135, 1228, 11069, 99642, 896803, 8071256, 72641337, 653772070, 5883948671, 52955538084, 476599842805, 4289398585298, 38604587267739, 347441285409712, 3126971568687473, 28142744118187326, 253284697063686007, 2279562273573174140, 20516060462158567341

Offset: 0

N. J. A. Sloane, Simon Plouffe

The o.g.f. of this sequence enabled the analysis of A162008, A162009 and A162010. - Johannes W. Meijer, Jun 27 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
G. Viennot, Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

From Johannes W. Meijer, Jun 27 2009: (Start)
Equals the second right hand column of triangle A162005 divided by 2.
Cf. A162008, A162009, A162010, A162011 and A162014 [2*(1+3*z)].
(End)

LinearRecurrence[{11, -19, 9}, {0, 1, 14}, 100] (* G. C. Greubel, Jul 06 2016 *)
Table[(3^(2 n + 1) - 8 n - 3)/16, {n, 0, 24}] (* Michael De Vlieger, Jul 08 2016 *)

G.f.: -x*(1+3*x)/(9*x-1)/(x-1)^2. - Simon Plouffe in his 1992 dissertation.
a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3). - Johannes W. Meijer, Jun 27 2009
a(n) = a(n-1) + (3^(2*n-1) - 1)/2. - Lechoslaw Ratajczak, Jul 06 2016
E.g.f.: (-3 - 8*x + 3*exp(8*x))*exp(x)/16. - Ilya Gutkovskiy, Jul 07 2016

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

Original entry on oeis.org

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

A162008 Third right hand column of the EG1 triangle A162005.

Original entry on oeis.org

16, 1032, 36096, 1035088, 27426960, 702812568, 17753262208, 445736371872, 11162877175440, 279268061007400, 6983654996144256, 174610650719469552, 4365455001524490256, 109138210900706764728, 2728473030627812279040

Offset: 1

Johannes W. Meijer, Jun 27 2009

Index entries for linear recurrences with constant coefficients, signature (46,-663,3748,-7711,6606,-2025).

Third right hand column of the EG1 triangle A162005.
Other right hand columns are A004004 (2x), A162009 and A162010.
Cf. A162011 and A162014.

a(n) = (64*n^2+112*n+30-432*n*9^n-405*9^n+375*25^n)/128.
a(n) = 46*a(n-1)-663*a(n-2)+3748*a(n-3)-7711*a(n-4)+6606*a(n-5)-2025*a(n-6).
G.f.: (16+296*z-768*z^2-1080*z^3)/((1-z)^3*(1-9*z)^2*(1-25*z)).

A162009 Fourth right hand column of the EG1 triangle A162005.

Original entry on oeis.org

272, 52736, 4766048, 319830400, 18598875760, 1002968825344, 51882638754240, 2621627565515520, 130715075544000720, 6468157990602644480, 318685706549526508832, 15663443488266952501376, 768809642314801857986608

Offset: 1

Johannes W. Meijer, Jun 27 2009

Fourth right hand column of the EG1 triangle A162005.
Other right hand columns are A004004 (2x), A162008 and A162010.
Cf. A162011 and A162014.

a(n) = (252105*49^n-525+76545*9^n-328125*25^n-225000*25^n*n-2320*n+134136*9^n*n-2112*n^2+46656*9^n*n^2-512*n^3)/3072
a(n) = 130*a(n-1)-6501*a(n-2)+163160*a(n-3)-2236466*a(n-4)+17123340*a(n-5)-71497186*a(n-6)+154127320*a(n-7)-174334221*a(n-8)+98986050*a(n-9)-22325625*a(n-10)
GF(z) = (272+17376*z-321360*z^2-1298624*z^3+8914800*z^4-11262240*z^5-10206000*z^6)/((1-z)^4*(1-9*z)^3*(1-25*z)^2*(1-49*z))

A162010 Fifth right hand column of the EG1 triangle A162005.

Original entry on oeis.org

7936, 3646208, 704357760, 93989648000, 10324483102720, 1013356176688128, 92857038223998720, 8148225153293502720, 695389790665420312320, 58282750219059501633280, 4827428305286309709508736

Offset: 1

Johannes W. Meijer, Jun 27 2009

Fifth right hand column of the EG1 triangle A162005.
Other right hand columns are A004004 (2x), A162008 and A162009.
Cf. A162011 and A162014.

a(n) = (13230+75008*n^2+65184*n-778248135*49^n+30720*n^3+ 502211745*81^n+ 295312500*25^n-26034048*9^n*n^2-43600032*9^n*n-395300640*49^n*n-19289340*9^n+ 4096*n^4+ 352500000*25^n*n+90000000*25^n*n^2-4478976*9^n*n^3)/98304
a(n)= 295*a(n-1)-36729*a(n-2)+2549775*a(n-3)-109746165*a(n-4)+3080128275*a(n-5)-57713313405*a(n-6)+727045264875*a(n-7)-6122436806115*a(n-8)+33837597147925*a(n-9)-119061300168619*a(n-10)+257794693911405*a(n-11)-339251103039591*a(n-12)+264193039731825*a(n-13)-112000136889375*a(n-14)+19937341265625*a(n-15)
GF(z) = (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)/(1-z)^5/(1-9*z)^4/(1-25*z)^3/(1-49*z)^2/(1-81*z)

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A004004 a(n) = (3^(2*n+1) - 8*n - 3)/16.

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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.

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A162008 Third right hand column of the EG1 triangle A162005.

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A162009 Fourth right hand column of the EG1 triangle A162005.

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A162010 Fifth right hand column of the EG1 triangle A162005.

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A004004 a(n) = (3^(2n+1) - 8n - 3)/16.