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A102413
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Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).
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7
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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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The n-sunlet graph is 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.
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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FORMULA
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G.f.: G(t,z) = (1 + t*z^2) / (1 - (1+t)*z - t*z^2).
The n-th row polynomial R(n,t) = [z^n] G(z,t)^n, where G(z,t) = 1/2*( 1 + (1 + t)*z + sqrt(1 + 2*(1 + t)*z + (1 + 6*t + t^2)*z^2) ).
exp( Sum_{n >= 1} R(n,t)*z^n/n ) = 1 + (1 + t)*z + (1 + 3*t + t^2)*z^2 + (1 + 5*t + 5*t^2 + t^3)*z^3 + ... is the o.g.f for A008288 read as a triangular array. (End)
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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;
Rows as polynomials:
1
1 + x,
1 + 4*x + x^2,
1 + 6*x + 6*x^2 + x^3,
1 + 8*x + 16*x^2 + 8*x^3 + x^4,
1 + 10*x + 30*x^2 + 30*x^3 + 10*x^4 + x^5,
... (End)
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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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MATHEMATICA
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CoefficientList[Table[2^-n ((1 + x - Sqrt[1 + x (6 + x)])^n + (1 + x + Sqrt[1 + x (6 + x)])^n), {n, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
LinearRecurrence[{1 + x, x}, {1, 1 + x, 1 + 4 x + x^2}, 10] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
Join[{1}, CoefficientList[CoefficientList[Series[(-1 - x - 2 x z)/(-1 + z + x z + x z^2), {z, 0, 10}], z], x]] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
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PROG
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(Haskell)
a102413 n k = a102413_tabl !! n !! k
a102413_row n = a102413_tabl !! n
a102413_tabl = [1] : [1, 1] : f [2] [1, 1] where
f us vs = ws : f vs ws where
ws = zipWith3 (((+) .) . (+))
([0] ++ us ++ [0]) ([0] ++ vs) (vs ++ [0])
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Row 0 in polynomials and Mathematica programs added by Georg Fischer, Apr 01 2019
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STATUS
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approved
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