A238360
Number of genus-10 rooted maps with n edges.
Original entry on oeis.org
15230046989184655753125, 5199629454143883380553750, 909887917857275652461097750, 108861830345440643086946970900, 10021124647635764856828690342402, 757187906770815991999545249667404, 48918614114003431712044170834572688, 2779227352989564224315657269511192976, 141720718575991991799057452443438430580
Offset: 20
Rooted maps with n edges of genus g for 0 <= g <= 10:
A000168,
A006300,
A006301,
A104742,
A215402,
A238355,
A238356,
A238357,
A238358,
A238359, this sequence.
-
T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 10];
Table[a[n], {n, 20, 30}] (* Jean-François Alcover, Jul 20 2018 *)
-
\\ see A238396
A238396
Triangle T(n,k) read by rows: T(n,k) is the number of rooted genus-k maps with n edges, n>=0, 0<=k<=n.
Original entry on oeis.org
1, 2, 0, 9, 1, 0, 54, 20, 0, 0, 378, 307, 21, 0, 0, 2916, 4280, 966, 0, 0, 0, 24057, 56914, 27954, 1485, 0, 0, 0, 208494, 736568, 650076, 113256, 0, 0, 0, 0, 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0, 17399772, 117822512, 248371380, 167808024, 24635754, 0, 0, 0, 0, 0, 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0
Offset: 0
Triangle starts:
00: 1,
01: 2, 0,
02: 9, 1, 0,
03: 54, 20, 0, 0,
04: 378, 307, 21, 0, 0,
05: 2916, 4280, 966, 0, 0, 0,
06: 24057, 56914, 27954, 1485, 0, 0, 0,
07: 208494, 736568, 650076, 113256, 0, 0, 0, 0,
08: 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0,
09: 17399772, 117822512, 248371380, 167808024, 24635754, 0, ...,
10: 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0, ...,
11: 1602117468, 18210135416, 73231116024, 117593590752, 66519597474, 8608033980, 0, ...,
12: 15792300756, 224636864830, 1183803697278, 2675326679856, 2416610807964, 672868675017, 24325703325, 0, ...,
...
- David M. Jackson and Terry I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman & Hall/CRC, circa 2000. See page 227.
Columns k for 0<=k<=10 are:
A000168,
A006300,
A006301,
A104742,
A215402,
A238355,
A238356,
A238357,
A238358,
A238359,
A238360.
See
A267180 for nonorientable analog.
The triangle without the zeros is
A269919.
-
T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4n - 2)/3 T[n-1, g] + (2n-3)(2n-2)(2n-1)/12 T[n-2, g-1] + 1/2 Sum[(2k-1)(2(n - k)-1) T[k-1, i] T[n-k-1, g-i] , {k, 1, n-1}, {i, 0, g}])/((n+1)/6);
Table[T[n, g], {n, 0, 10}, {g, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018, after Gheorghe Coserea *)
-
N=20;
MEM=matrix(N+1,N+1, r,c, -1); \\ for memoization
Q(n,g)=
{
if (n<0, return( (g<=0) ) ); \\ not given in paper
if (g<0, return( 0 ) ); \\ not given in paper
if (n<=0, return( g==0 ) ); \\ as in paper
my( m = MEM[n+1,g+1] );
if ( m != -1, return(m) ); \\ memoized value
my( t=0 );
t += (4*n-2)/3 * Q(n-1, g);
t += (2*n-3)*(2*n-2)*(2*n-1)/12 * Q(n-2, g-1);
my(l, j);
t += 1/2*
sum(k=1, n-1, l=n-k; \\ l+k == n, both >= 1
sum(i=0, g, j=g-i; \\ i+j == g, both >= 0
(2*k-1)*(2*l-1) * Q(k-1, i) * Q(l-1, j)
);
);
t *= 6/(n+1);
MEM[n+1, g+1] = t; \\ memoize
return(t);
}
for (n=0, N, for (g=0, n, print1(Q(n, g),", "); ); print(); ); /* print triangle */
A239921
Number of unrooted maps with n edges of (orientable) genus 7.
Original entry on oeis.org
508233789579, 104295987346126, 11269592389125547, 852994611088758224, 50777879440443305426, 2531246455428148382456, 109880399953287962099588, 4265557888300762164284822, 150940131172496245801920542, 4938911033961317567088755908, 151101665358744941452325232448
Offset: 14
A348800
a(n) = number of unsensed genus 7 maps with n edges.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254118439668, 52148049818864, 5634797561708385, 426497331688178676, 25388940147173859412, 1265623233919838264624, 54940200059090328012148
Offset: 0