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A235114 Irregular triangle read by rows: T(n,k) = number of independent vertex subsets of size k of the graph g_n obtained by attaching two pendant edges to each vertex of the star graph S_n (having n vertices). 1
1, 1, 3, 1, 1, 6, 10, 6, 1, 1, 9, 28, 40, 28, 9, 1, 1, 12, 55, 129, 170, 129, 55, 12, 1, 1, 15, 91, 300, 597, 748, 597, 300, 91, 15, 1, 1, 18, 136, 580, 1560, 2783, 3368, 2783, 1560, 580, 136, 18, 1, 1, 21, 190, 996, 3391, 7923, 13067, 15418, 13067, 7923, 3391, 996, 190, 21, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Sum of entries in row n = 4*5^{n-1} + 4^{n-1} (n>=1) (see A235115).

In the Maple program P[n] gives the independence polynomial of the graph g_n.

REFERENCES

E. Mandrescu, Unimodality of some independence polynomials via their palindromicity, Australasian J. of Combinatorics, 53, 2012, 77-82.

D. Stevanovic, Graphs with palindromic independence polynomial, Graph Theory Notes of New  York, 34, 1998, 31-36.

LINKS

Table of n, a(n) for n=0..63.

FORMULA

Generating polynomial of row n (n>=1) is x(1 + x)^{2n-2} + (1 + x)^2*(1 + 3*x + x^2)^{n-1}  (it is palindromic).

Bivariate generating polynomial: G(x,z) = (1 - z - 2xz - x^2*z - x^2*z^2)/((1 - z - 2xz - x^2*z)(1 - z - 3xz - x^2*z)).

G(1/x, x^2*z) = G(x,z) (this implies the above mentioned palindromicity).

EXAMPLE

Row 1 is 1,3,1; indeed, S_1 is the one-vertex graph and after attaching two pendant vertices we obtain the path graph ABC; the independent vertex subsets are: empty, {A}, {B}, {C}, and {A, C}.

Triangle begins:

1;

1,3,1;

1,6,10,6,1;

1,9,28,40,28,9,1;

MAPLE

G := (1-z-2*x*z-x^2*z-x^2*z^2)/((1-z-2*x*z-x^2*z)*(1-z-3*x*z-x^2*z)): Gser := simplify(series(G, z = 0, 10)): for n from 0 to 9 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 9 do seq(coeff(P[n], x, i), i = 0 .. 2*n) end do; # yields sequence in triangular form

CROSSREFS

Cf. A235115.

Sequence in context: A120247 A235113 A235116 * A272866 A228899 A255918

Adjacent sequences:  A235111 A235112 A235113 * A235115 A235116 A235117

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jan 13 2014

STATUS

approved

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Last modified February 20 06:26 EST 2019. Contains 320332 sequences. (Running on oeis4.)