A143084 -id:A143084 - cp's OEIS frontend

A100732 a(n) = (3*n)!.

1, 6, 720, 362880, 479001600, 1307674368000, 6402373705728000, 51090942171709440000, 620448401733239439360000, 10888869450418352160768000000, 265252859812191058636308480000000

Offset: 0

Ralf Stephan, Dec 08 2004

a(n) equals (-1)^n times the determinant of the (3n+1) X (3n+1) matrix with consecutive integers from 1 to 3n+1 along the main diagonal, consecutive integers from 2 to 3n+1 along the superdiagonal, consecutive integers from 1 to 3n along the subdiagonal, and 1's everywhere else (see Mathematica code below). - John M. Campbell, Jul 12 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000142, A010050, A100733, A100734, A143084, A143819, A268504.

a100732 = a000142 . a008585  -- Reinhard Zumkeller, Feb 19 2013

[Factorial(3*n): n in [0..15]]; // Vincenzo Librandi, Sep 24 2011

Table[(-1)^n*Det[Array[KroneckerDelta[#1, #2]*(#1 - 1) + KroneckerDelta[#1, #2 - 1]*(#1) + KroneckerDelta[#1, #2 + 1]*(#1 - 2) + 1 &, {3*n + 1, 3*n + 1}]], {n, 0, 24}] (* John M. Campbell, Jul 12 2011 *)
(3Range[0,10])! (* Harvey P. Dale, Sep 23 2011 *)

[factorial(3*n) for n in range(0, 11)] # Peter Luschny, Jun 06 2016

a(n) = A000142(A008585(n)).
E.g.f.: 1/(1-x^3).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(2*Pi)*3^(3*n+1/2)*n^(3*n+1/2)/exp(3*n).
Sum_{n>=0} 1/a(n) = (exp(3/2) + 2*cos(sqrt(3)/2))/(3*exp(1/2)) = A143819. (End)
Sum_{n>=0} (-1)^n/a(n) = (1 + 2*exp(3/2)*cos(sqrt(3)/2))/(3*e). - Amiram Eldar, Feb 14 2021
a(n) = A143084(2n,n). - Alois P. Heinz, Jul 12 2024

A374574 a(n) = Sum_{j=n..2n} j!.

Original entry on oeis.org

1, 3, 32, 870, 46224, 4037880, 522956160, 93928267440, 22324392518400, 6780385526302080, 2561327494111411200, 1177652997443424902400, 647478071469567800985600, 419450149241406188889984000, 316196664211373618844934963200, 274410818470142134209609852672000

Offset: 0

Alois P. Heinz, Jul 11 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..224

Row sums of A143084.
Cf. A000142, A100822, A143122, A296591 (the same for product).
Diagonal of A054115, A211370.

a:= proc(n) option remember; `if`(n<3, [1, 3, 32][n+1],
     ((16*n^3-16*n^2-n+2)*a(n-1)-(n-1)*(16*n^3-20*n^2+6*n-1)
      *a(n-2)+2*(2*n-1)*(4*n+1)*(n-1)*(n-2)*a(n-3))/(4*n-3))
    end:
seq(a(n), n=0..15);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1) -(n-1)! +(2*n-1)! +(2*n)!)
    end:
seq(a(n), n=0..15);

a(n) = a(n-1) - (n-1)! + (2*n-1)! + (2*n)! with a(0) = 1.
a(n) = Sum_{j=0..n} (n + j)!.
a(n) = A100822(2n,n).
a(n) = A143122(2n,n).

A370418 Triangle read by rows. T(n, k) = (n - k)! * (n + k)!.

Original entry on oeis.org

1, 1, 2, 4, 6, 24, 36, 48, 120, 720, 576, 720, 1440, 5040, 40320, 14400, 17280, 30240, 80640, 362880, 3628800, 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600, 25401600, 29030400, 43545600, 87091200, 239500800, 958003200, 6227020800, 87178291200

Offset: 0

Peter Luschny, Feb 27 2024

			Triangle starts:
[0]      1;
[1]      1,      2;
[2]      4,      6,     24;
[3]     36,     48,    120,     720;
[4]    576,    720,   1440,    5040,   40320;
[5]  14400,  17280,  30240,   80640,  362880,  3628800;
[6] 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600;

Cf. A010050 (main diagonal), A009445 (subdiagonal), A001044 (column 0), A175430 (column 1), A024420 (bisection is alternating sum).

Cf. A119502, A143084.

T := (n, k) -> (n - k)! * (n + k)!:
seq(seq(T(n, k), k = 0..n), n = 0..7);

Table[(n - k)!*(n + k)!, {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 05 2024 *)

Sum_{k=0..n} (-1)^k*T(n, k) = n!^2 / 2 + (-1)^n * (2*n + 2)! / (2*n + 2)^2.

cp's OEIS Frontend

A100732 a(n) = (3*n)!.

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A374574 a(n) = Sum_{j=n..2n} j!.

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A370418 Triangle read by rows. T(n, k) = (n - k)! * (n + k)!.

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