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Irregular triangle read by rows: coefficients in order of decreasing exponents of polynomials P_g(x) related to Hultman numbers.
3

%I #40 May 31 2023 09:51:00

%S 1,1,12,8,1,72,528,704,180,1,324,8760,53792,98124,56160,8064,1,1344,

%T 103040,1759520,9936360,21676144,19083456,6356160,604800,1,5436,

%U 1054056,41312704,539233128,2901894144,7118351104,8247838464,4418632656,988952832,68428800,1,21816,10106736,823376896,21574613676,235937470944,1230387808384,3281254260864,4608240745104,3390175943424,1247151098880,204083712000,10897286400

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

%C Row n contains 2*n-1 terms.

%C Evaluating the polynomials at 1 gives A035319.

%H Gheorghe Coserea, <a href="/A185259/b185259.txt">Rows n=1..101, flattened</a>

%H Nikita Alexeev and Peter Zograf, <a href="http://arxiv.org/abs/1111.3061">Hultman numbers, polygon gluings and matrix integrals</a>, arXiv preprint arXiv:1111.3061 [math.PR], 2011.

%H Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 9.

%e Triangle begins:

%e [1] 1

%e [2] 1 12 8

%e [3] 1 72 528 704 180

%e [4] 1 324 8760 53792 98124 56160 8064

%e [5] 1 1344 103040 1759520 9936360 21676144 19083456 6356160 604800

%e [6] ...

%t 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);

%t row[n_] := CoefficientList[P[n, x], x] // Reverse;

%t Array[row, 7] // Flatten (* _Jean-François Alcover_, Sep 05 2018, after _Gheorghe Coserea_ *)

%o (PARI)

%o P(n, v='x) = {

%o my(x='x+O('x^(2*n-1)), f=(1-x)^(4*n+1),

%o s=sum(k=0, 2*n-2, -stirling(2*n+2+k, k+1, 1)/binomial(2*n+2+k,2)*x^k));

%o subst(Pol(f*s, 'x), 'x, v);

%o };

%o concat(vector(7, n, Vec(P(n))))

%o \\ test: N=50; vector(N, n, P(n,1)) == vector(N, n, (4*n)!/((2*n+1)!*4^n))

%o \\ _Gheorghe Coserea_, Jan 30 2018

%Y Cf. A035319, A185263.

%K nonn,tabf

%O 1,3

%A _N. J. A. Sloane_, Jan 21 2012

%E More terms from _Gheorghe Coserea_, Jan 30 2018