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
Examples
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] ...
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.
Programs
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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 *) -
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
Extensions
More terms from Gheorghe Coserea, Jan 30 2018
Comments