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A284947 Irregular triangle read by rows: coefficients of the cycle polynomial of the n-complete graph K_n.

%I A284947 #12 Feb 16 2025 08:33:43
%S A284947 0,0,0,1,0,0,0,4,3,0,0,0,10,15,12,0,0,0,20,45,72,60,0,0,0,35,105,252,
%T A284947 420,360,0,0,0,56,210,672,1680,2880,2520,0,0,0,84,378,1512,5040,12960,
%U A284947 22680,20160,0,0,0,120,630,3024,12600,43200,113400,201600,181440
%N A284947 Irregular triangle read by rows: coefficients of the cycle polynomial of the n-complete graph K_n.
%H A284947 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>
%H A284947 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclePolynomial.html">Cycle Polynomial</a>
%F A284947 T(n, k) = binomial(n, k)*Pochhammer(3, k-3) if k >= 3 else 0. - _Peter Luschny_, Oct 22 2017
%e A284947 1: 0
%e A284947 2: 0
%e A284947 3: x^3
%e A284947 4: x^3 (4 + 3 x)
%e A284947 5: x^3 (10 + 15 x + 12 x^2)
%e A284947 6: x^3 (20 + 45 x + 72 x^2 + 60 x^3)
%e A284947 giving
%e A284947 1 3-cycle in K_3
%e A284947 4 3-cycles and 3 4-cycles in K_4
%e A284947 From _Peter Luschny_, Oct 22 2017: (Start)
%e A284947 Prepending six zeros leads to the regular triangle:
%e A284947 [0] 0
%e A284947 [1] 0, 0
%e A284947 [2] 0, 0, 0
%e A284947 [3] 0, 0, 0,  1
%e A284947 [4] 0, 0, 0,  4,   3
%e A284947 [5] 0, 0, 0, 10,  15,   12
%e A284947 [6] 0, 0, 0, 20,  45,   72,   60
%e A284947 [7] 0, 0, 0, 35, 105,  252,  420,   360
%e A284947 [8] 0, 0, 0, 56, 210,  672, 1680,  2880,  2520
%e A284947 [9] 0, 0, 0, 84, 378, 1512, 5040, 12960, 22680, 20160
%e A284947 (End)
%p A284947 A284947row := n -> seq(`if`(k<3, 0, pochhammer(3,k-3)*binomial(n,k)), k=0..n):
%p A284947 seq(A284947row(n), n=3..10); # _Peter Luschny_, Oct 22 2017
%t A284947 CoefficientList[Table[-(n*x*(2 - x + n*x - 2*HypergeometricPFQ[{1, 1, 1 - n}, {2}, -x]))/4, {n, 10}], x] // Flatten
%Y A284947 Cf. A002807 (row sums of a(n)).
%Y A284947 Cf. A144151 (generalization to include 1- and 2-"cycles").
%K A284947 nonn,tabf
%O A284947 3,8
%A A284947 _Eric W. Weisstein_, Apr 06 2017