A102413 - OEIS

A102413

Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).

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, and Pierre-Jean Spaenlehauer, A Polyhedral Method for Sparse Systems with many Positive Solutions, arXiv:1804.05683 [math.CO], 2018.

A. F. Horadam, Chebyshev and Pell connections, Fib. Quart. 43 (2) (2005) 108-121, table (6.11)

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] F(z, t)^n, where F(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)

From Peter Bala, Aug 01 2024: (Start)

T(n, k) = A008288(n-k, k) + A008288(n-k-1, k-1) (Bihan et al., Proposition 6.6).

T(n, k) = 1 if n = 0 or k = n, else for 1 <= k <= n-1, T(n, k) = Sum_{j = 0..min(n-k, k)} (2^j)*(binomial(n-k, j)*binomial(k, j) + binomial(n-k-1, j)*binomial(k-1, j)).

Let S(x) = (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x) denote the g.f. of the sequence of large Schröder numbers A006318. The signed n-th row polynomial R(n, -x) = 1/S(x)^n + (x*S(x))^n. (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, 4, 1;

1, 6, 6, 1;

1, 8, 16, 8, 1;

From Eric W. Weisstein, Apr 03 2018: (Start)

Rows as polynomials:

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, A006318, A008288.

Cf. A241023 (central terms).