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A235117 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 ladder graph L_n (i.e., L_n is the 2 X n grid graph; 0 <= k <= 4n+1).

%I A235117 #16 Feb 27 2020 22:38:53
%S A235117 1,1,6,10,6,1,1,12,54,124,162,124,54,12,1,1,18,134,556,1451,2530,3036,
%T A235117 2530,1451,556,134,18,1,1,24,250,1516,6042,16892,34430,52352,60122,
%U A235117 52352,34430,16892,6042,1516,250,24,1,1,30,402,3220,17393,67676,197588,444584,784702,1098826,1228500,1098826,784702,444584,197588,67676,17393,3220,402,30,1
%N A235117 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 ladder graph L_n (i.e., L_n is the 2 X n grid graph; 0 <= k <= 4n+1).
%C A235117 Each row is palindromic (see the Stevanovic and Mandrescu references).
%C A235117 Row n (n >= 0) contains 4n+1 entries.
%C A235117 Sum of entries in row n = A235118(n).
%C A235117 In the Maple program, P(n) gives the independence polynomial of the graph g_n.
%H A235117 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 A235117 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 A235117 Generating polynomial P(n) of row n (i.e., the independence polynomial of the graph g_n) satisfies the recurrence relation P(n) = (1 + x)^2*(1 + 3x + x^2)P(n-1) + x(1 + x)^6 *P(n-2); P(0)=1; P(1)=(1 + 4x + x^2)*(1 + x)^2.
%e A235117 Triangle begins:
%e A235117 1;
%e A235117 1,6,10,6,1;
%e A235117 1,12,54,124,162,124,54,12,1;
%e A235117 1,18,134,556,1451,2530,3036,2530,1451,556,134,18,1;
%p A235117 P := proc (n) option remember: if n = 0 then 1 elif n = 1 then sort(expand((1+x)^2*(1+4*x+x^2))) else sort(expand((1+x)^2*(1+3*x+x^2)*P(n-1) +x*(1+x)^6*P(n-2))) end if end proc: for n from 0 to 5 do seq(coeff(P(n), x, i), i = 0 .. 4*n) end do; # yields sequence in triangular form
%Y A235117 Cf. A235118.
%K A235117 nonn,tabf
%O A235117 0,3
%A A235117 _Emeric Deutsch_, Jan 14 2014