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A094021
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Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k components (1<=k<=n).
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3
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1, 1, 1, 3, 3, 1, 12, 14, 6, 1, 55, 75, 40, 10, 1, 273, 429, 275, 90, 15, 1, 1428, 2548, 1911, 770, 175, 21, 1, 7752, 15504, 13328, 6370, 1820, 308, 28, 1, 43263, 95931, 93024, 51408, 17640, 3822, 504, 36, 1, 246675, 600875, 648945, 406980, 162792, 42840
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OFFSET
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1,4
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LINKS
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FORMULA
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T(n, k) = binomial(n, k-1)*binomial(3n-2k-1, n-k)/(2n-k).
G.f.: G=G(t, z) satisfies G^3+(t^3*z^2-t^2*z-3)G^2+(t^2*z+3)G-1=0.
O.g.f. A(x,t) = revert( x/((1 + x*t)*C(x)) ) with respect to x, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f for the Catalan numbers A000108.
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EXAMPLE
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Triangle begins:
1;
1, 1;
3, 3, 1;
12, 14, 6, 1;
55, 75, 40, 10, 1;
273, 429, 275, 90, 15, 1;
1428, 2548, 1911, 770, 175, 21, 1;
7752, 15504, 13328, 6370, 1820, 308, 28, 1;
(End)
T(3,2)=3 because, with A,B,C denoting the vertices of a triangle, we have the 2-component forests (A,BC), (B,CA) and (C,AB).
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MAPLE
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T:=proc(n, k) if k<=n then binomial(n, k-1)*binomial(3*n-2*k-1, n-k)/(2*n-k) else 0 fi end: seq(seq(T(n, k), k=1..n), n=1..11);
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MATHEMATICA
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T[n_, k_] := If[k <= n, Binomial[n, k-1]*Binomial[3n-2k-1, n-k]/(2n-k), 0];
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PROG
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(PARI)
T(n, k)=binomial(n, k-1)*binomial(3*n-2*k-1, n-k)/(2*n-k);
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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