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A321710
Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.
8
1, 3, 12, 1, 56, 15, 288, 165, 8, 1584, 1611, 252, 9152, 14805, 4956, 180, 54912, 131307, 77992, 9132, 339456, 1138261, 1074564, 268980, 8064, 2149888, 9713835, 13545216, 6010220, 579744, 13891584, 81968469, 160174960, 112868844, 23235300, 604800, 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880, 608583680, 5702382933, 19588944336, 28540603884, 16497874380, 2936606400, 68428800, 4107939840, 47168678571, 206254571236, 404562365316, 344901105444, 108502598960, 8099018496
OFFSET
1,2
COMMENTS
Row n contains floor((n+1)/2) = A008619(n-1) terms.
LINKS
Gheorghe Coserea, Rows n = 1..42, flattened
Alain Giorgetti and Timothy R. S. Walsh, Enumeration of hypermaps of a given genus, Ars Math. Contemp. 15 (2018) 225-266.
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
P. G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.
FORMULA
A000257(n)=T(n,0), A118093(n)=T(n,1), A214817(n)=T(n,2), A214818(n)=T(n,3), A060593(n)=T(2*n+1,n)=(2*n)!/(n+1), A003319(n+1)=Sum_{k=0..floor((n-1)/2)} T(n,k).
EXAMPLE
Triangle starts:
n\k [0] [1] [2] [3] [4] [5]
[1] 1;
[2] 3;
[3] 12, 1;
[4] 56, 15;
[5] 288, 165, 8;
[6] 1584, 1611, 252;
[7] 9152, 14805, 4956, 180;
[8] 54912, 131307, 77992, 9132;
[9] 339456, 1138261, 1074564, 268980, 8064;
[10] 2149888, 9713835, 13545216, 6010220, 579744;
[11] 13891584, 81968469, 160174960, 112868844, 23235300, 604800;
[12] 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880;
[13] ...
MATHEMATICA
l1[f_, n_] := Sum[(i-1)t[i]D[f, t[i-1]], {i, 2, n}];
m1[f_, n_] := Sum[(i-1)t[j]t[i-j]D[f, t[i-1]] + j(i-j)t[i+1]D[f, t[j], t[i-j]], {i, 2, n}, {j, i-1}];
ff[1] = x^2 t[1];
ff[n_] := ff[n] = Simplify@(2x l1[ff[n-1], n] + m1[ff[n-1], n] + Sum[t[i+1]j(i-j)D[ff[k], t[j]]D[ff[n-1-k], t[i-j]], {i, 2, n-1}, {j, i-1}, {k, n-2}]) / n;
row[n_]:=Reverse[CoefficientList[n ff[n] /. {t[_]->x}, x]][[;;;; 2]][[;; Quotient[n+1, 2]]];
Table[row[n], {n, 14}] (* Andrei Zabolotskii, Jun 27 2025, after the PARI code *)
PROG
(PARI)
L1(f, N) = sum(i=2, N, (i-1)*t[i]*deriv(f, t[i-1]));
M1(f, N) = {
sum(i=2, N, sum(j=1, i-1, (i-1)*t[j]*t[i-j]*deriv(f, t[i-1]) +
j*(i-j)*t[i+1]*deriv(deriv(f, t[j]), t[i-j])));
};
F(N) = {
my(u='x, v='x, f=vector(N)); t=vector(N+1, n, eval(Str("t", n)));
f[1] = u*v*t[1];
for (n=2, N, f[n] = (u + v)*L1(f[n-1], n) + M1(f[n-1], n) +
sum(i=2, n-1, t[i+1]*sum(j=1, i-1,
j*(i-j)*sum(k=1, n-2, deriv(f[k], t[j])*deriv(f[n-1-k], t[i-j]))));
f[n] /= n);
f;
};
seq(N) = {
my(f=F(N), v=substvec(f, t, vector(#t, n, 'x)),
g=vector(#v, n, Polrev(Vec(n * v[n]))));
apply(p->Vecrev(substpol(p, 'x^2, 'x)), g);
};
concat(seq(14))
CROSSREFS
Columns k=0..9 give: A000257 (k=0), A118093 (k=1), A214817 (k=2), A214818 (k=3), A318104 (k=4), A321705 (k=5), A321706 (k=6), A321707 (k=7), A321708 (k=8), A321709 (k=9).
Row sums give A003319(n+1).
Sequence in context: A162854 A342787 A110121 * A383131 A358325 A288518
KEYWORD
nonn,tabf
AUTHOR
Gheorghe Coserea, Nov 17 2018
STATUS
approved