|
%I #23 Dec 17 2018 19:08:44
%S 1,3,12,1,56,15,288,165,8,1584,1611,252,9152,14805,4956,180,54912,
%T 131307,77992,9132,339456,1138261,1074564,268980,8064,2149888,9713835,
%U 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 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.
%C Row n contains floor((n+1)/2) = A008619(n-1) terms.
%H Gheorghe Coserea, <a href="/A321710/b321710.txt">Rows n = 1..42, flattened</a>
%H 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 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 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 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 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 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 Triangle starts:
%e n\k [0] [1] [2] [3] [4] [5]
%e [1] 1;
%e [2] 3;
%e [3] 12, 1;
%e [4] 56, 15;
%e [5] 288, 165, 8;
%e [6] 1584, 1611, 252;
%e [7] 9152, 14805, 4956, 180;
%e [8] 54912, 131307, 77992, 9132;
%e [9] 339456, 1138261, 1074564, 268980, 8064;
%e [10] 2149888, 9713835, 13545216, 6010220, 579744;
%e [11] 13891584, 81968469, 160174960, 112868844, 23235300, 604800;
%e [12] 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880;
%e [13] ...
%o (PARI)
%o L1(f, N) = sum(i=2, N, (i-1)*t[i]*deriv(f, t[i-1]));
%o M1(f, N) = {
%o sum(i=2, N, sum(j=1, i-1, (i-1)*t[j]*t[i-j]*deriv(f, t[i-1]) +
%o j*(i-j)*t[i+1]*deriv(deriv(f, t[j]), t[i-j])));
%o };
%o F(N) = {
%o my(u='x, v='x, f=vector(N)); t=vector(N+1, n, eval(Str("t", n)));
%o f[1] = u*v*t[1];
%o for (n=2, N, f[n] = (u + v)*L1(f[n-1], n) + M1(f[n-1], n) +
%o sum(i=2, n-1, t[i+1]*sum(j=1, i-1,
%o j*(i-j)*sum(k=1, n-2, deriv(f[k], t[j])*deriv(f[n-1-k], t[i-j]))));
%o f[n] /= n);
%o f;
%o };
%o seq(N) = {
%o my(f=F(N), v=substvec(f, t, vector(#t, n, 'x)),
%o g=vector(#v, n, Polrev(Vec(n * v[n]))));
%o apply(p->Vecrev(substpol(p, 'x^2, 'x)), g);
%o };
%o concat(seq(14))
%Y 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 Row sums give A003319(n+1).
%Y Cf. A008619, A060593.
%K nonn,tabf
%O 1,2
%A _Gheorghe Coserea_, Nov 17 2018
|
