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

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, 420, 360, 0, 0, 0, 56, 210, 672, 1680, 2880, 2520, 0, 0, 0, 84, 378, 1512, 5040, 12960, 22680, 20160, 0, 0, 0, 120, 630, 3024, 12600, 43200, 113400, 201600, 181440

Offset: 3

Author

Eric W. Weisstein, Apr 06 2017

Keywords

• nonn
• tabf

Examples

			1: 0
2: 0
3: x^3
4: x^3 (4 + 3 x)
5: x^3 (10 + 15 x + 12 x^2)
6: x^3 (20 + 45 x + 72 x^2 + 60 x^3)
giving
1 3-cycle in K_3
4 3-cycles and 3 4-cycles in K_4
From _Peter Luschny_, Oct 22 2017: (Start)
Prepending six zeros leads to the regular triangle:
[0] 0
[1] 0, 0
[2] 0, 0, 0
[3] 0, 0, 0,  1
[4] 0, 0, 0,  4,   3
[5] 0, 0, 0, 10,  15,   12
[6] 0, 0, 0, 20,  45,   72,   60
[7] 0, 0, 0, 35, 105,  252,  420,   360
[8] 0, 0, 0, 56, 210,  672, 1680,  2880,  2520
[9] 0, 0, 0, 84, 378, 1512, 5040, 12960, 22680, 20160
(End)
		

Links

• Eric Weisstein's World of Mathematics, Complete Graph
• Eric Weisstein's World of Mathematics, Cycle Polynomial

Crossrefs

Cf. A002807 (row sums of a(n)).

Cf. A144151 (generalization to include 1- and 2-"cycles").

Programs

Maple

A284947row := n -> seq(`if`(k<3, 0, pochhammer(3,k-3)*binomial(n,k)), k=0..n):
seq(A284947row(n), n=3..10); # Peter Luschny, Oct 22 2017

Mathematica

CoefficientList[Table[-(n*x*(2 - x + n*x - 2*HypergeometricPFQ[{1, 1, 1 - n}, {2}, -x]))/4, {n, 10}], x] // Flatten

Formula

T(n, k) = binomial(n, k)*Pochhammer(3, k-3) if k >= 3 else 0. - Peter Luschny, Oct 22 2017