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A143359
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Triangle read by rows, T(n,k) = number of symmetric ordered trees with n edges and root degree k (1<=k<=n).
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1
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1, 1, 1, 2, 0, 1, 3, 1, 1, 1, 6, 0, 3, 0, 1, 10, 2, 4, 2, 1, 1, 20, 0, 10, 0, 4, 0, 1, 35, 5, 15, 5, 5, 3, 1, 1, 70, 0, 35, 0, 15, 0, 5, 0, 1, 126, 14, 56, 14, 21, 9, 6, 4, 1, 1, 252, 0, 126, 0, 56, 0, 21, 0, 6, 0, 1, 462, 42, 210, 42, 84, 28, 28, 14, 7, 5, 1, 1, 924, 0, 462, 0, 210, 0, 84, 0, 28
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Sum of entries in row n is binom(n,floor(n/2)) (A001405).
T(n,1)=A001405(n-1).
Sum(k*T(n,k),k=1..n)=A143360(n).
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FORMULA
| G.f.=(1+tzS)/[1-t^2*z^2*C(z^2)]-1, where S = 1/[1-z-z^2*C(z^2)] is the g.f. of the sequence binom(n, floor(n/2)) (A001405) and C(z)=[1-sqrt(1-4z)]/(2z) is the generating function of the Catalan numbers (A000108).
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EXAMPLE
| Triangle starts:
1;
1,1;
2,0,1;
3,1,1,1;
6,0,3,0,1;
10,2,4,2,1,1;
20,0,10,0,4,0,1;
35,5,15,5,5,3,1,1;
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MAPLE
| C:=proc(z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: S:=1/(1-z-z^2*C(z^2)): G:=(1+t*z*S)/(1-t^2*z^2*C(z^2))-1: Gser:=simplify(series(G, z=0, 15)): for n to 13 do P[n]:=coeff(Gser, z, n) end do: for n to 13 do seq(coeff(P[n], t, j), j=1..n) end do; # yields sequence in triangular form
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CROSSREFS
| Cf. A001405, A000108, A143360.
Sequence in context: A094184 A078805 A122837 * A130504 A044942 A114912
Adjacent sequences: A143356 A143357 A143358 * A143360 A143361 A143362
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 15 2008
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