cp's OEIS Frontend

A308244 Triangle T(n,k) read by rows, giving even-numbered coefficients of the matching polynomial of the n-ladder graph.

%I A308244 #16 Jan 25 2024 14:31:13
%S A308244 1,-1,1,2,-4,1,-3,11,-7,1,5,-26,29,-10,1,-8,56,-94,56,-13,1,13,-114,
%T A308244 263,-234,92,-16,1,-21,223,-667,815,-473,137,-19,1,34,-424,1577,-2504,
%U A308244 1982,-838,191,-22,1,-55,789,-3538,7018,-7191,4115,-1356,254,-25,1,89,-1444,7622,-18336,23431,-17266
%N A308244 Triangle T(n,k) read by rows, giving even-numbered coefficients of the matching polynomial of the n-ladder graph.
%C A308244 For 0 <= k <= n, T(n,k) is the coefficient of x^(2*k) in the matching polynomial of the n-ladder graph.  We take T(0,0)=1.
%H A308244 Robert Israel, <a href="/A308244/b308244.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)
%H A308244 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>.
%H A308244 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MatchingPolynomial.html">Matching Polynomial</a>.
%F A308244 Generating function as triangle: (1+x)/(1+2*x-x*y+x^2*y-x^3).
%F A308244 T(n,k) = T(n-1,k-1)-2*T(n-1,k)-T(n-2,k-1)+T(n-3,k) for n >= 3 (taking T(n,k)=0 unless 0 <= k <= n).
%F A308244 T(n,0) = (-1)^n*A000045(n+1).
%F A308244 T(n,1) = (-1)^(n+1)*A002940(n) for n >= 1.
%F A308244 T(n,2) = (-1)^n*A002941(n-1) for n >= 2.
%e A308244 Triangle begins
%e A308244 1
%e A308244 -1 1
%e A308244 2 -4 1
%e A308244 -3 11 -7 1
%e A308244 5 -26 29 -10 1
%e A308244 -8 56 -94 56 -13 1
%e A308244 13 -114 263 -234 92 -16 1
%e A308244 -21 223 -667 815 -473 137 -19 1
%e A308244 34 -424 1577 -2504 1982 -838 191 -22 1
%p A308244 g:= gfun:-rectoproc({a(n+3)+(-x^2+2)*a(n+2)+x^2*a(n+1)-a(n),a(0)=1,a(1)=x^2-1,a(2)=x^4-4*x^2+2}, a(n), remember):
%p A308244 for nn from 0 to 10 do
%p A308244 seq(coeff(g(nn),x,k),k=0..2*nn,2)
%p A308244 od;
%Y A308244 Cf. A000045, A002940, A002941.
%K A308244 sign,tabl
%O A308244 0,4
%A A308244 _Robert Israel_, May 16 2019