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 A102413 Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n). 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 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). 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 Frédéric Bihan, Francisco Santos, Pierre-Jean Spaenlehauer, A Polyhedral Method for Sparse Systems with many Positive Solutions, arXiv:1804.05683 [math.CO], 2018. Eric Weisstein's World of Mathematics, Matching-Generating Polynomial Eric Weisstein's World of Mathematics, Sunlet Graph 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; From Eric W. Weisstein, Apr 03 2018: (Start) 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) 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 MATHEMATICA 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 *) 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 EXTENSIONS Row 0 in polynomials and Mathematica programs added by Georg Fischer, Apr 01 2019 STATUS approved

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Last modified June 4 17:50 EDT 2020. Contains 334828 sequences. (Running on oeis4.)