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A102413 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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row n contains n+1 terms. Row sums yield A099425. T(n,k) = T(n,n-k).

For n > 2: same recurrence like A008288 and A128966. - Reinhard Zumkeller, Apr 15 2014

REFERENCES

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.

LINKS

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

FORMULA

G.f.: G(t,z) = (1 + t*z^2) / (1 - (1+t)*z - t*z^2).

For n > 2: T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1), 0 < k < n. - Reinhard Zumkeller, Apr 15 2014 (corrected by Andrew Woods, Dec 08 2014)

From Peter Bala, Jun 25 2015: (Start)

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)

EXAMPLE

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;

MAPLE

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

PROG

(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])

-- Reinhard Zumkeller, Apr 15 2014

CROSSREFS

Cf. A099425, A008288.

Cf. A241023 (central terms).

Sequence in context: A132046 A141540 A143188 * A144480 A144463 A174376

Adjacent sequences:  A102410 A102411 A102412 * A102414 A102415 A102416

KEYWORD

nonn,tabl,easy

AUTHOR

Emeric Deutsch, Jan 07 2005

STATUS

approved

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Last modified December 7 03:36 EST 2016. Contains 278838 sequences.