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A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts. 7
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 (list; graph; refs; listen; history; text; internal format)
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.

Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps

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] ...

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).

Cf. A008619, A060593.

Sequence in context: A162854 A342787 A110121 * A358325 A288518 A069522

Adjacent sequences: A321707 A321708 A321709 * A321711 A321712 A321713

KEYWORD

nonn,tabf

AUTHOR

Gheorghe Coserea, Nov 17 2018

STATUS

approved

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Last modified February 1 14:29 EST 2023. Contains 359993 sequences. (Running on oeis4.)