A238359
Number of genus-9 rooted maps with n edges.
Original entry on oeis.org
11665426077721040625, 3498878057690404966500, 540996834819906946713375, 57494374008560749302297480, 4724172886681078698955547790, 320061005837218582787265273000, 18618409220753939214291224549409, 956146512935178711199035220384800, 44232688287025023758415781081779828, 1871678026675570344184400604255444240
Offset: 18
Rooted maps with n edges of genus g for 0 <= g <= 10:
A000168,
A006300,
A006301,
A104742,
A215402,
A238355,
A238356,
A238357,
A238358, this sequence,
A238360.
-
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, 9];
Table[a[n], {n, 18, 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 */
A239924
Number of unrooted maps with n edges of (orientable) genus 10.
Original entry on oeis.org
380751174738424280720, 123800701289478148878890, 20679270860399513431761798, 2366561529248819497695971912, 208773430159079852919281433050, 15143758135416335992767275804168, 940742579115450773885994408739386
Offset: 20