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A302232 Triangle T(n,k) of the numbers of k-matchings in the n-Moebius ladder (0 <= k <= n, n > 2).

%I A302232 #12 Jun 02 2025 15:21:19
%S A302232 1,9,18,6,1,12,42,44,7,1,15,75,145,95,13,1,18,117,336,420,192,18,1,21,
%T A302232 168,644,1225,1085,371,31,1,24,228,1096,2834,3880,2588,696,47,1,27,
%U A302232 297,1719,5652,10656,11097,5823,1278,78,1,30,375,2540,10165,24626,35645,29380,12535,2310,123
%N A302232 Triangle T(n,k) of the numbers of k-matchings in the n-Moebius ladder (0 <= k <= n, n > 2).
%C A302232 Initial terms in each row match those in A061702.
%H A302232 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Matching-GeneratingPolynomial.html">Matching-Generating Polynomial</a>
%H A302232 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusLadder.html">Moebius Ladder</a>
%F A302232 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))).
%F A302232 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).
%e A302232 As polynomials sum(k=0..n) x^k*T(n, k):
%e A302232 1 + 9*x + 18*x^2 + 6*x^3,
%e A302232 1 + 12*x + 42*x^2 + 44*x^3 + 7*x^4,
%e A302232 1 + 15*x + 75*x^2 + 145*x^3 + 95*x^4 + 13*x^5,
%e A302232 1 + 18*x + 117*x^2 + 336*x^3 + 420*x^4 + 192*x^5 + 18*x^6,
%e A302232 ...
%t A302232 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
%t A302232 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
%Y A302232 Row sums are A020877.
%Y A302232 Cf. A061702.
%K A302232 nonn,tabl
%O A302232 3,2
%A A302232 _Eric W. Weisstein_, Apr 03 2018