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A185259
Irregular triangle read by rows: coefficients in order of decreasing exponents of polynomials P_g(x) related to Hultman numbers.
3
1, 1, 12, 8, 1, 72, 528, 704, 180, 1, 324, 8760, 53792, 98124, 56160, 8064, 1, 1344, 103040, 1759520, 9936360, 21676144, 19083456, 6356160, 604800, 1, 5436, 1054056, 41312704, 539233128, 2901894144, 7118351104, 8247838464, 4418632656, 988952832, 68428800, 1, 21816, 10106736, 823376896, 21574613676, 235937470944, 1230387808384, 3281254260864, 4608240745104, 3390175943424, 1247151098880, 204083712000, 10897286400
OFFSET
1,3
COMMENTS
Row n contains 2*n-1 terms.
Evaluating the polynomials at 1 gives A035319.
LINKS
Gheorghe Coserea, Rows n=1..101, flattened
Nikita Alexeev and Peter Zograf, Hultman numbers, polygon gluings and matrix integrals, arXiv preprint arXiv:1111.3061 [math.PR], 2011.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 9.
EXAMPLE
Triangle begins:
[1] 1
[2] 1 12 8
[3] 1 72 528 704 180
[4] 1 324 8760 53792 98124 56160 8064
[5] 1 1344 103040 1759520 9936360 21676144 19083456 6356160 604800
[6] ...
MATHEMATICA
P[n_, x_] := (f = (1-x)^(4n+1); s = Sum[-StirlingS1[2n+2+k, k+1]/ Binomial[2n+2+k, 2] x^k, {k, 0, 2n-2}]; f s + O[x]^(2n-1) // Normal);
row[n_] := CoefficientList[P[n, x], x] // Reverse;
Array[row, 7] // Flatten (* Jean-François Alcover, Sep 05 2018, after Gheorghe Coserea *)
PROG
(PARI)
P(n, v='x) = {
my(x='x+O('x^(2*n-1)), f=(1-x)^(4*n+1),
s=sum(k=0, 2*n-2, -stirling(2*n+2+k, k+1, 1)/binomial(2*n+2+k, 2)*x^k));
subst(Pol(f*s, 'x), 'x, v);
};
concat(vector(7, n, Vec(P(n))))
\\ test: N=50; vector(N, n, P(n, 1)) == vector(N, n, (4*n)!/((2*n+1)!*4^n))
\\ Gheorghe Coserea, Jan 30 2018
CROSSREFS
Sequence in context: A206423 A319406 A038333 * A090438 A346223 A128108
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jan 21 2012
EXTENSIONS
More terms from Gheorghe Coserea, Jan 30 2018
STATUS
approved