|
%I #11 Oct 23 2017 22:25:34
%S 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 420,360,0,0,0,56,210,672,1680,2880,2520,0,0,0,84,378,1512,5040,12960,
%U 22680,20160,0,0,0,120,630,3024,12600,43200,113400,201600,181440
%N Irregular triangle read by rows: coefficients of the cycle polynomial of the n-complete graph K_n.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteGraph.html">Complete 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)*Pochhammer(3, k-3) if k >= 3 else 0. - _Peter Luschny_, Oct 22 2017
%e 1: 0
%e 2: 0
%e 3: x^3
%e 4: x^3 (4 + 3 x)
%e 5: x^3 (10 + 15 x + 12 x^2)
%e 6: x^3 (20 + 45 x + 72 x^2 + 60 x^3)
%e giving
%e 1 3-cycle in K_3
%e 4 3-cycles and 3 4-cycles in K_4
%e From _Peter Luschny_, Oct 22 2017: (Start)
%e Prepending six zeros leads to the regular triangle:
%e [0] 0
%e [1] 0, 0
%e [2] 0, 0, 0
%e [3] 0, 0, 0, 1
%e [4] 0, 0, 0, 4, 3
%e [5] 0, 0, 0, 10, 15, 12
%e [6] 0, 0, 0, 20, 45, 72, 60
%e [7] 0, 0, 0, 35, 105, 252, 420, 360
%e [8] 0, 0, 0, 56, 210, 672, 1680, 2880, 2520
%e [9] 0, 0, 0, 84, 378, 1512, 5040, 12960, 22680, 20160
%e (End)
%p A284947row := n -> seq(`if`(k<3, 0, pochhammer(3,k-3)*binomial(n,k)), k=0..n):
%p seq(A284947row(n), n=3..10); # _Peter Luschny_, Oct 22 2017
%t CoefficientList[Table[-(n*x*(2 - x + n*x - 2*HypergeometricPFQ[{1, 1, 1 - n}, {2}, -x]))/4, {n, 10}], x] // Flatten
%Y Cf. A002807 (row sums of a(n)).
%Y Cf. A144151 (generalization to include 1- and 2-"cycles").
%K nonn,tabf
%O 3,8
%A _Eric W. Weisstein_, Apr 06 2017
|
