%I #12 Dec 10 2023 18:11:05
%S 1,1,1,3,1,4,1,1,5,4,1,6,8,1,1,7,13,5,1,8,19,13,1,1,9,26,26,6,1,10,34,
%T 45,19,1,1,11,43,71,45,7,1,12,53,105,90,26,1,1,13,64,148,161,71,8,1,
%U 14,76,201,266,161,34,1,1,15,89,265,414,322,105,9,1,16,103,341
%N Triangle T(n,k) of the numbers of k-matchings in the n-pan graph (0 <= k <= ceiling(n/2)).
%C The n-th row gives the coefficients of the matching-generating polynomial of the n-pan graph.
%C Sequence extended to a(1)-a(2) using the formula/recurrence.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching-GeneratingPolynomial.html">Matching-Generating Polynomial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PanGraph.html">Pan Graph</a>
%F Sum_{k=0..ceiling(n/2)} T(n,k)*x^k = x^(n/2)*(sqrt(x)*Fibonacci(n, 1/sqrt(x)) + lucasl(n, 1/sqrt(x))).
%e Rows as matching-generating polynomials:
%e 1 + x,
%e 1 + 3*x,
%e 1 + 4*x + x^2,
%e 1 + 5*x + 4*x^2,
%e 1 + 6*x + 8*x^2 + x^3,
%e 1 + 7*x + 13*x^2 + 5*x^3,
%e 1 + 8*x + 19*x^2 + 13*x^3 + x^4,
%e 1 + 9*x + 26*x^2 + 26*x^3 + 6*x^4,
%e ...
%t CoefficientList[Table[x^(n/2) (Sqrt[x] Fibonacci[n, 1/Sqrt[x]] + LucasL[n, 1/Sqrt[x]]), {n, 20}], x] // Flatten
%t CoefficientList[LinearRecurrence[{1, x}, {1 + x, 1 + 3 x}, 20], x] // Flatten
%t CoefficientList[CoefficientList[Series[-(( 1 + x + 2 x z)/(-1 + z + x z^2)), {z, 0, 20}], z], x] // Flatten
%K nonn,tabf
%O 1,4
%A _Eric W. Weisstein_, Apr 03 2018