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%I #31 Sep 07 2022 04:06:38
%S 0,0,1,0,9,6,0,36,96,72,0,100,600,1800,1440,0,225,2400,16200,51840,
%T 43200,0,441,7350,88200,635040,2116800,1814400,0,784,18816,352800,
%U 4515840,33868800,116121600,101606400,0,1296,42336,1143072,22861440,304819200,2351462400,8230118400,7315660800
%N 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 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 Pontus von Brömssen, <a href="/A291909/b291909.txt">Rows n = 1..100, flattened</a> (rows n = 1..60 from Vincenzo Librandi).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclePolynomial.html">Cycle Polynomial</a>
%F T(n, k) = binomial(n, k)^2*k!*(k - 1)!/2 for k > 1.
%e Cycle polynomials are
%e 0
%e x^4
%e 9 x^4 + 6 x^6
%e 36 x^4 + 96 x^6 + 72 x^8
%e 100 x^4 + 600 x^6 + 1800 x^8 + 1440 x^10
%e ...
%e so the first few rows are
%e 0;
%e 0, 1;
%e 0, 9, 6;
%e 0, 36, 96, 72;
%e ...
%t CoefficientList[Table[Sum[Binomial[n, k]^2 k! (k - 1)! x^k, {k, 2, n}]/2, {n, 10}], x] // Flatten
%t 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 (PARI) T(n, k) = if(k>1, binomial(n, k)^2*k!*(k - 1)!/2, 0) \\ _Andrew Howroyd_, Apr 29 2018
%Y Cf. A070968 (row sums), A010796 (main diagonal).
%K nonn,tabl
%O 1,5
%A _Eric W. Weisstein_, Sep 05 2017
%E Terms T(n,0) for n >= 3 deleted (in order to have a regular triangle) by _Pontus von Brömssen_, Sep 06 2022
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