OFFSET
1,4
COMMENTS
The n-th row gives the coefficients of the matching-generating polynomial of the n-pan graph.
Sequence extended to a(1)-a(2) using the formula/recurrence.
LINKS
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Pan Graph
FORMULA
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))).
EXAMPLE
Rows as matching-generating polynomials:
1 + x,
1 + 3*x,
1 + 4*x + x^2,
1 + 5*x + 4*x^2,
1 + 6*x + 8*x^2 + x^3,
1 + 7*x + 13*x^2 + 5*x^3,
1 + 8*x + 19*x^2 + 13*x^3 + x^4,
1 + 9*x + 26*x^2 + 26*x^3 + 6*x^4,
...
MATHEMATICA
CoefficientList[Table[x^(n/2) (Sqrt[x] Fibonacci[n, 1/Sqrt[x]] + LucasL[n, 1/Sqrt[x]]), {n, 20}], x] // Flatten
CoefficientList[LinearRecurrence[{1, x}, {1 + x, 1 + 3 x}, 20], x] // Flatten
CoefficientList[CoefficientList[Series[-(( 1 + x + 2 x z)/(-1 + z + x z^2)), {z, 0, 20}], z], x] // Flatten
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Eric W. Weisstein, Apr 03 2018
STATUS
approved