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A271715
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Number of set partitions of [3n] with minimal block length multiplicity equal to n.
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3
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1, 4, 55, 1540, 67375, 4239235, 383563180, 51925673800, 10652498631775, 3139051466175625, 1228555090548911125, 602267334323068414000, 357161594247065690582500, 250870551734754490461422500, 205672479804595549379158525000, 194557626586812183102927448930000
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OFFSET
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0,2
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LINKS
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FORMULA
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Recursion: see Maple program.
For n>0, a(n) = (3^n + n!)*(3*n)! / (6^n * (n!)^2). - Vaclav Kotesovec, Apr 16 2016
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MAPLE
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a:= proc(n) option remember; `if`(n<5,
[1, 4, 55, 1540, 67375][n+1], ((2*(3*n-2))*
(3*n-1)*(n^2-n-9)*a(n-1) -(3*(n-3))*(3*n-1)*
(3*n-4)*(3*n-2)*(3*n-5)*a(n-2))/(4*n*(n-4)))
end:
seq(a(n), n=0..20);
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MATHEMATICA
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multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i&, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]] // Union}]]];
a[n_] := If[n==0, 1, b[3n, 3n, n] - b[3n, 3n, n+1]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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