A185259 Irregular triangle read by rows: coefficients in order of decreasing exponents of polynomials P_g(x) related to Hultman numbers.

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

Cf. A035319, A185263.

Sequence in context: A206423 A319406 A038333 * A090438 A346223 A128108

Adjacent sequences: A185256 A185257 A185258 * A185260 A185261 A185262

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jan 21 2012

Extensions

More terms from Gheorghe Coserea, Jan 30 2018

Status

approved