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

3,8

Links

Table of n, a(n) for n=3..62.

Eric Weisstein's World of Mathematics, Complete Graph

Eric Weisstein's World of Mathematics, Cycle Polynomial

Formula

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

Example

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)

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

Crossrefs

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

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

Sequence in context: A309528 A293496 A290326 * A261099 A030120 A058878

Adjacent sequences: A284944 A284945 A284946 * A284948 A284949 A284950

Keyword

nonn,tabf

Author

Eric W. Weisstein, Apr 06 2017

Status

approved