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

1, -1, 1, 2, -4, 1, -3, 11, -7, 1, 5, -26, 29, -10, 1, -8, 56, -94, 56, -13, 1, 13, -114, 263, -234, 92, -16, 1, -21, 223, -667, 815, -473, 137, -19, 1, 34, -424, 1577, -2504, 1982, -838, 191, -22, 1, -55, 789, -3538, 7018, -7191, 4115, -1356, 254, -25, 1, 89, -1444, 7622, -18336, 23431, -17266

Offset: 0

Author

Robert Israel, May 16 2019

Keywords

• sign
• tabl

Comments

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.

Examples

			Triangle begins
1
-1 1
2 -4 1
-3 11 -7 1
5 -26 29 -10 1
-8 56 -94 56 -13 1
13 -114 263 -234 92 -16 1
-21 223 -667 815 -473 137 -19 1
34 -424 1577 -2504 1982 -838 191 -22 1
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
• Eric Weisstein's World of Mathematics, Ladder Graph.
• Eric Weisstein's World of Mathematics, Matching Polynomial.

Crossrefs

Cf. A000045, A002940, A002941.

Programs

Maple

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):
for nn from 0 to 10 do
seq(coeff(g(nn),x,k),k=0..2*nn,2)
od;

Formula

Generating function as triangle: (1+x)/(1+2*x-x*y+x^2*y-x^3).

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).

T(n,0) = (-1)^n*A000045(n+1).

T(n,1) = (-1)^(n+1)*A002940(n) for n >= 1.

T(n,2) = (-1)^n*A002941(n-1) for n >= 2.