A302232 Triangle T(n,k) of the numbers of k-matchings in the n-Moebius ladder (0 <= k <= n, n > 2).
1, 9, 18, 6, 1, 12, 42, 44, 7, 1, 15, 75, 145, 95, 13, 1, 18, 117, 336, 420, 192, 18, 1, 21, 168, 644, 1225, 1085, 371, 31, 1, 24, 228, 1096, 2834, 3880, 2588, 696, 47, 1, 27, 297, 1719, 5652, 10656, 11097, 5823, 1278, 78, 1, 30, 375, 2540, 10165, 24626, 35645, 29380, 12535, 2310, 123
Offset: 3
Examples
As polynomials sum(k=0..n) x^k*T(n, k): 1 + 9*x + 18*x^2 + 6*x^3, 1 + 12*x + 42*x^2 + 44*x^3 + 7*x^4, 1 + 15*x + 75*x^2 + 145*x^3 + 95*x^4 + 13*x^5, 1 + 18*x + 117*x^2 + 336*x^3 + 420*x^4 + 192*x^5 + 18*x^6, ...
Links
- Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
- Eric Weisstein's World of Mathematics, Moebius Ladder
Programs
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Mathematica
CoefficientList[LinearRecurrence[{1 + x, 2 x (1 + x), -(-1 + x) x^2, -x^4}, {1 + 3 x, 1 + 6 x + 3 x^2, 1 + 9 x + 18 x^2 + 6 x^3, 1 + 12 x + 42 x^2 + 44 x^3 + 7 x^4}, {3, 10}], x] // Flatten CoefficientList[CoefficientList[Series[-((-1 - 9 x - 18 x^2 - 6 x^3 - 2 x z - 15 x^2 z - 20 x^3 z - x^4 z - x^2 z^2 - 5 x^3 z^2 + 4 x^4 z^2 + 6 x^5 z^2 + x^4 z^3 + 6 x^5 z^3 + 3 x^6 z^3)/((1 + x z) (1 - z - 2 x z - x z^2 + x^3 z^3))), {z, 0, 10}], z], x] // Flatten
Formula
G.f.: -((z^2*(-1 - 9*x - 18*x^2 - 6*x^3 - 2*x*z - 15*x^2*z - 20*x^3*z - x^4*z - x^2*z^2 - 5*x^3*z^2 + 4*x^4*z^2 + 6*x^5*z^2 + x^4*z^3 + 6*x^5*z^3 + 3*x^6*z^3))/((1 + x*z)*(1 - z - 2*x*z - x*z^2 + x^3*z^3))).
Writing t(n, x) = sum(k=0..n) x^k*T(n, k), t(n, x) == (1 + x)*t(n-1, x) + 2*x*(1 + x)*t(n-2, x) -(-1 + x)*x^2*t(n-3, x) -x^4*t(n-4, x).
Comments