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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 (list; graph; refs; listen; history; text; internal format)
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: A073275 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

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Last modified November 23 00:33 EST 2017. Contains 295107 sequences.