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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).

%I A235114 #10 Feb 27 2020 14:46:37
%S A235114 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,
%T A235114 91,300,597,748,597,300,91,15,1,1,18,136,580,1560,2783,3368,2783,1560,
%U A235114 580,136,18,1,1,21,190,996,3391,7923,13067,15418,13067,7923,3391,996,190,21,1
%N 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).
%C A235114 Sum of entries in row n = 4*5^{n-1} + 4^{n-1} (n>=1) (see A235115).
%C A235114 In the Maple program P[n] gives the independence polynomial of the graph g_n.
%H A235114 E. Mandrescu, <a href="http://ajc.maths.uq.edu.au/pdf/53/ajc_v53_p077.pdf">Unimodality of some independence polynomials via their palindromicity</a>, Australasian J. of Combinatorics, 53, 2012, 77-82.
%H A235114 D. Stevanovic, <a href="https://draganstevanovic.files.wordpress.com/2012/09/gtn-341998-31-36-acro6.pdf">Graphs with palindromic independence polynomial</a>, Graph Theory Notes of New York, 34, 1998, 31-36.
%F A235114 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).
%F A235114 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)).
%F A235114 G(1/x, x^2*z) = G(x,z) (this implies the above mentioned palindromicity).
%e A235114 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}.
%e A235114 Triangle begins:
%e A235114 1;
%e A235114 1,3,1;
%e A235114 1,6,10,6,1;
%e A235114 1,9,28,40,28,9,1;
%p A235114 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
%Y A235114 Cf. A235115.
%K A235114 nonn,tabf
%O A235114 0,3
%A A235114 _Emeric Deutsch_, Jan 13 2014