A361030 a(n) = 20160*(3*n)!/(n!*(n+3)!^2).

560, 210, 504, 2352, 15840, 135135, 1361360, 15519504, 194699232, 2636552100, 38003792400, 577037174400, 9155656500480, 150853746558690, 2568167588473200, 44990491457326800, 808333317429976800, 14853124707775823700, 278470827854627007600, 5316261259042879236000

Offset

0,1

Comments

Row 2 of square array A361027.

The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 60*A000984(n) is divisible by (n + 1)*(n + 2)*(n + 3) and the result 60*(2*n)!/(n!*(n+3)!) is the super ballot number A007272(n). Similarly, the de Bruijn numbers A006480(n) = (3*n)!/n!^3 have the property that 20160*A006480(n) is divisible by ((n + 1)*(n + 2)*(n + 3))^2.

Equivalently, the central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that (1*3*5)*A000984(n+3) is divisible by (2*n + 1)*(2*n + 3)*(2*n + 5). The result is always an even integer. In fact, (1/2)*(1*3*5)/((2*n + 1)*(2n + 3)*(2n + 5))*A000984(n+3) = A007272(n).

Similarly, the de Bruijn numbers A006480(n) = (3*n)!/n!^3 have the property that (1*2*4*5*7*8)*A006480(n+3) is divisible by (3*n + 1)*(3*n + 2)*(3*n + 4)*(3*n + 5)*(3*n + 7)*(3*n + 8). The result is always an integer divisible by 3.

Links

Table of n, a(n) for n=0..19.

Peter Bala, A361030 is an integer sequence

Formula

a(n) = 20160/((n+1)*(n+2)*(n+3))^2 * (3*n)!/n!^3.

a(n) = (1/3)*(1*2*4*5*7*8) * A006480(n+3)/((3*n + 1)*(3*n + 2)*(3*n + 4)*

(3*n + 5)*(3*n + 7)*(3*n + 8)), where A006480(n) = (3*n)!/n!^3.

a(n) = (1/5)*A007272(n)*A361038(n). Using this it can be shown that a(n) is always an integer.

a(n) = (1/3)*27^(n+3)*binomial(7/3, n+3)*binomial(8/3, n+3).

a(n) ~ sqrt(3)*10080*(27^n)/(Pi*n^7).

P-recursive: (n + 3)^2*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 560.

The o.g.f. A(x) satisfies the differential equation x^2*(1 - 27*x)*A''(x) + x*(7 - 54*x)*A'(x) + (9 - 6*x)*A(x) - 5040 = 0, with A(0) = 560 and A'(0) = 210.

Maple

a := proc(n) option remember; if n = 0 then 560 else 3*(3*n-1)*(3*n-2)/(n+3)^2*a(n-1) end if; end proc:
seq(a(n), n = 0..20);

Crossrefs

Row 2 of A361027. Cf. A006480, A007272, A361028, A361029, A361031, A361038.

Sequence in context: A350664 A050254 A139197 * A136346 A234263 A100987

Adjacent sequences: A361027 A361028 A361029 * A361031 A361032 A361033

Keyword

nonn,easy

Author

Peter Bala, Mar 01 2023

Status

approved