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A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.

%I A321710 #27 Jun 27 2025 16:28:36
%S A321710 1,3,12,1,56,15,288,165,8,1584,1611,252,9152,14805,4956,180,54912,
%T A321710 131307,77992,9132,339456,1138261,1074564,268980,8064,2149888,9713835,
%U A321710 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
%N A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.
%C A321710 Row n contains floor((n+1)/2) = A008619(n-1) terms.
%H A321710 Gheorghe Coserea, <a href="/A321710/b321710.txt">Rows n = 1..42, flattened</a>
%H A321710 Alain Giorgetti and Timothy R. S. Walsh, <a href="https://amc-journal.eu/index.php/amc/article/download/1115/1221">Enumeration of hypermaps of a given genus</a>, Ars Math. Contemp. 15 (2018) 225-266.
%H A321710 Timothy R. Walsh, <a href="http://www.info2.uqam.ca/~walsh_t/papers/GENERATING NONISOMORPHIC.pdf">Space-efficient generation of nonisomorphic maps and hypermaps</a>
%H A321710 T. R. Walsh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Walsh/walsh3.html">Space-Efficient Generation of Nonisomorphic Maps and Hypermaps</a>, J. Int. Seq. 18 (2015) # 15.4.3.
%H A321710 P. G. Zograf, <a href="https://doi.org/10.1093/imrn/rnv077">Enumeration of Grothendieck's Dessins and KP Hierarchy</a>, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
%H A321710 Peter Zograf, <a href="https://arxiv.org/abs/1312.2538">Enumeration of Grothendieck's Dessins and KP Hierarchy</a>, arXiv:1312.2538 [math.CO], 2014.
%F A321710 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).
%e A321710 Triangle starts:
%e A321710 n\k  [0]       [1]        [2]         [3]         [4]        [5]
%e A321710 [1]  1;
%e A321710 [2]  3;
%e A321710 [3]  12,       1;
%e A321710 [4]  56,       15;
%e A321710 [5]  288,      165,       8;
%e A321710 [6]  1584,     1611,      252;
%e A321710 [7]  9152,     14805,     4956,       180;
%e A321710 [8]  54912,    131307,    77992,      9132;
%e A321710 [9]  339456,   1138261,   1074564,    268980,     8064;
%e A321710 [10] 2149888,  9713835,   13545216,   6010220,    579744;
%e A321710 [11] 13891584, 81968469,  160174960,  112868844,  23235300,  604800;
%e A321710 [12] 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880;
%e A321710 [13] ...
%t A321710 l1[f_,n_] := Sum[(i-1)t[i]D[f,t[i-1]], {i,2,n}];
%t A321710 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}];
%t A321710 ff[1] = x^2 t[1];
%t A321710 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;
%t A321710 row[n_]:=Reverse[CoefficientList[n ff[n] /. {t[_]->x}, x]][[;;;;2]][[;;Quotient[n+1,2]]];
%t A321710 Table[row[n], {n,14}] (* _Andrei Zabolotskii_, Jun 27 2025, after the PARI code *)
%o A321710 (PARI)
%o A321710 L1(f, N) = sum(i=2, N, (i-1)*t[i]*deriv(f, t[i-1]));
%o A321710 M1(f, N) = {
%o A321710   sum(i=2, N, sum(j=1, i-1, (i-1)*t[j]*t[i-j]*deriv(f, t[i-1]) +
%o A321710       j*(i-j)*t[i+1]*deriv(deriv(f, t[j]), t[i-j])));
%o A321710 };
%o A321710 F(N) = {
%o A321710   my(u='x, v='x, f=vector(N)); t=vector(N+1, n, eval(Str("t", n)));
%o A321710   f[1] = u*v*t[1];
%o A321710   for (n=2, N, f[n] = (u + v)*L1(f[n-1], n) + M1(f[n-1], n) +
%o A321710     sum(i=2, n-1, t[i+1]*sum(j=1, i-1,
%o A321710     j*(i-j)*sum(k=1, n-2, deriv(f[k], t[j])*deriv(f[n-1-k], t[i-j]))));
%o A321710     f[n] /= n);
%o A321710   f;
%o A321710 };
%o A321710 seq(N) = {
%o A321710   my(f=F(N), v=substvec(f, t, vector(#t, n, 'x)),
%o A321710      g=vector(#v, n, Polrev(Vec(n * v[n]))));
%o A321710   apply(p->Vecrev(substpol(p, 'x^2, 'x)), g);
%o A321710 };
%o A321710 concat(seq(14))
%Y A321710 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).
%Y A321710 Row sums give A003319(n+1).
%Y A321710 Cf. A008619, A060593.
%K A321710 nonn,tabf
%O A321710 1,2
%A A321710 _Gheorghe Coserea_, Nov 17 2018