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A291909 Triangle read by rows: T(n,k) is the coefficient of x^(2*k) in the cycle polynomial of the complete bipartite graph K_{n,n}, 1 <= k <= n.

%I A291909 #32 Feb 16 2025 08:33:51
%S A291909 0,0,1,0,9,6,0,36,96,72,0,100,600,1800,1440,0,225,2400,16200,51840,
%T A291909 43200,0,441,7350,88200,635040,2116800,1814400,0,784,18816,352800,
%U A291909 4515840,33868800,116121600,101606400,0,1296,42336,1143072,22861440,304819200,2351462400,8230118400,7315660800
%N A291909 Triangle read by rows: T(n,k) is the coefficient of x^(2*k) in the cycle polynomial of the complete bipartite graph K_{n,n}, 1 <= k <= n.
%C A291909 Also the coefficients of x^(2*k) in the chordless cycle polynomial of the n X n rook graph. - _Eric W. Weisstein_, Feb 21 2018
%H A291909 Pontus von Brömssen, <a href="/A291909/b291909.txt">Rows n = 1..100, flattened</a> (rows n = 1..60 from Vincenzo Librandi).
%H A291909 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>
%H A291909 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A291909 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclePolynomial.html">Cycle Polynomial</a>
%F A291909 T(n, k) = binomial(n, k)^2*k!*(k - 1)!/2 for k > 1.
%e A291909 Cycle polynomials are
%e A291909         0
%e A291909       x^4
%e A291909     9 x^4 +   6 x^6
%e A291909    36 x^4 +  96 x^6 +   72 x^8
%e A291909   100 x^4 + 600 x^6 + 1800 x^8 + 1440 x^10
%e A291909   ...
%e A291909 so the first few rows are
%e A291909   0;
%e A291909   0,  1;
%e A291909   0,  9,  6;
%e A291909   0, 36, 96, 72;
%e A291909   ...
%t A291909 CoefficientList[Table[Sum[Binomial[n, k]^2 k! (k - 1)! x^k, {k, 2, n}]/2, {n, 10}], x] // Flatten
%t A291909 Join[{0}, CoefficientList[Table[n^2 (HypergeometricPFQ[{1, 1, 1 - n, 1 - n}, {2}, x] - 1)/2, {n, 2, 10}], x]] // Flatten (* _Eric W. Weisstein_, Feb 21 2018 *)
%o A291909 (PARI) T(n, k) = if(k>1, binomial(n, k)^2*k!*(k - 1)!/2, 0) \\ _Andrew Howroyd_, Apr 29 2018
%Y A291909 Cf. A070968 (row sums), A010796 (main diagonal).
%K A291909 nonn,tabl
%O A291909 1,5
%A A291909 _Eric W. Weisstein_, Sep 05 2017
%E A291909 Terms T(n,0) for n >= 3 deleted (in order to have a regular triangle) by _Pontus von Brömssen_, Sep 06 2022