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A291909 Triangle read by rows: coefficients of the terms x^(2k) in the cycle polynomials of the complete bipartite graph K_{n,n}. 2
0, 0, 1, 0, 0, 9, 6, 0, 0, 36, 96, 72, 0, 0, 100, 600, 1800, 1440, 0, 0, 225, 2400, 16200, 51840, 43200, 0, 0, 441, 7350, 88200, 635040, 2116800, 1814400, 0, 0, 784, 18816, 352800, 4515840, 33868800, 116121600, 101606400, 0, 0, 1296, 42336, 1143072, 22861440, 304819200, 2351462400, 8230118400, 7315660800, 0, 0, 2025, 86400, 3175200, 91445760, 1905120000, 26127360000, 205752960000, 731566080000, 658409472000 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Also the coefficients of x^(2 k) in the chordless cycle polynomial of the n X n rook graph. - Eric W. Weisstein, Feb 21 2018

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1888

Eric Weisstein's World of Mathematics, Chordless Cycle

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Eric Weisstein's World of Mathematics, Cycle Polynomial

FORMULA

T(n, k) = binomial(n, k)^2*k!*(k - 1)!/2 for k > 1.

EXAMPLE

Cycle polynomials are

        0

      x^4

    9 x^4 +   6 x^6

   36 x^4 +  96 x^6 +   72 x^8

  100 x^4 + 600 x^6 + 1800 x^8 + 1440 x^10

  ...

so the first few rows are

  0;

  0, 1;

  0, 0,  9,  6;

  0, 0, 36, 96, 72;

  ...

MATHEMATICA

CoefficientList[Table[Sum[Binomial[n, k]^2 k! (k - 1)! x^k, {k, 2, n}]/2, {n, 10}], x] // Flatten

Join[{0}, CoefficientList[Table[n^2 (HypergeometricPFQ[{1, 1, 1 - n, 1 - n}, {2}, x] - 1)/2, {n, 2, 10}], x]] // Flatten (* Eric W. Weisstein, Feb 21 2018 *)

PROG

(PARI) T(n, k) = if(k>1, binomial(n, k)^2*k!*(k - 1)!/2, 0) \\ Andrew Howroyd, Apr 29 2018

CROSSREFS

Row sums are A070968.

Sequence in context: A317869 A103362 A252245 * A263177 A154161 A130590

Adjacent sequences:  A291906 A291907 A291908 * A291910 A291911 A291912

KEYWORD

nonn,tabl

AUTHOR

Eric W. Weisstein, Sep 05 2017

STATUS

approved

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Last modified October 14 00:10 EDT 2019. Contains 327990 sequences. (Running on oeis4.)