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A102413
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Triangle read by rows: T(n,k) is the number of k-matchings of the corona C'(n) of the cycle graph C(n) and the complete graph K(1); in other words, C'(n) is the graph constructed from C(n) to which for each vertex v a new vertex v' and the edge vv' is added.
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2
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1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 16, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 48, 76, 48, 12, 1, 1, 14, 70, 154, 154, 70, 14, 1, 1, 16, 96, 272, 384, 272, 96, 16, 1, 1, 18, 126, 438, 810, 810, 438, 126, 18, 1, 1, 20, 160, 660, 1520, 2004, 1520, 660, 160, 20, 1, 1, 22, 198, 946, 2618, 4334, 4334, 2618, 946, 198, 22, 1
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OFFSET
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0,5
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COMMENTS
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Row n contains n+1 terms. Row sums yield A099425. T(n,k)=T(n,n-k)
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REFERENCES
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J. L. Gross and J. Yellen, Handbook of Graph Theory, CRC Press, Boca Raton, 2004, p. 894.
F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969, p. 167.
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LINKS
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Table of n, a(n) for n=0..77.
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FORMULA
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G.f.=G(t, z)=(1+tz^2)/[1-(1+t)z-tz^2].
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EXAMPLE
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T(3,2)=6 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following six 2-matchings: {Aa,BC},{Bb,AC},{Cc,AB},{Aa,Bb},{Aa,Cc} and {Bb,Cc}.
The triangle starts:
1;
1,1;
1,4,1;
1,6,6,1;
1,8,16,8,1;
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MAPLE
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G:=(1+t*z^2)/(1-(1+t)*z-t*z^2): Gser:=simplify(series(G, z=0, 38)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser, z^n) od:for n from 0 to 11 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A099425.
Sequence in context: A132046 A141540 A143188 * A144480 A144463 A174376
Adjacent sequences: A102410 A102411 A102412 * A102414 A102415 A102416
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch, Jan 07 2005
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STATUS
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approved
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