A302240 Triangle T(n,k) of the numbers of k-matchings in the n-pan graph (0 <= k <= ceiling(n/2)).

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, 45, 19, 1, 1, 11, 43, 71, 45, 7, 1, 12, 53, 105, 90, 26, 1, 1, 13, 64, 148, 161, 71, 8, 1, 14, 76, 201, 266, 161, 34, 1, 1, 15, 89, 265, 414, 322, 105, 9, 1, 16, 103, 341

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

Table of n, a(n) for n=1..74.

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

Sequence in context: A217780 A329316 A109411 * A130307 A130314 A334466

Adjacent sequences: A302237 A302238 A302239 * A302241 A302242 A302243

Keyword

nonn,tabf

Author

Eric W. Weisstein, Apr 03 2018

Status

approved