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A361030
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a(n) = 20160*(3*n)!/(n!*(n+3)!^2).
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6
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560, 210, 504, 2352, 15840, 135135, 1361360, 15519504, 194699232, 2636552100, 38003792400, 577037174400, 9155656500480, 150853746558690, 2568167588473200, 44990491457326800, 808333317429976800, 14853124707775823700, 278470827854627007600, 5316261259042879236000
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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