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A284947
Irregular triangle read by rows: coefficients of the cycle polynomial of the n-complete graph K_n.
2
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
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
KEYWORD
nonn,tabf
AUTHOR
Eric W. Weisstein, Apr 06 2017
STATUS
approved