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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. 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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.

LINKS

Joerg Arndt, Table of n, a(n) for n = 0..1325 (rows 0..50, flattened)

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).

FORMULA

From Gheorghe Coserea, Mar 11 2016: (Start)

(n+1)/6 * 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_{k=1..n-1} Sum_{i=0..g} (2*k-1) * (2*(n-k)-1) * T(k-1, i) * T(n-k-1, g-i) for all n >= 1 and 0 <= g <= n/2, with the initial conditions T(0,0) = 1 and T(n,g) = 0 for g < 0 or g > n/2.

For column g, as n goes to infinity we have T(n,g) ~ t(g) * n^(5*(g-1)/2) * 12^n, where t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)) and gamma is the Gamma function.

(End)

EXAMPLE

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, ...,

...

MATHEMATICA

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-Fran├žois Alcover, Jul 19 2018, after Gheorghe Coserea *)

PROG

(PARI) 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 */

CROSSREFS

Columns k for 0<=k<=10 are: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

Sum of row n is A000698(n+1).

See A267180 for nonorientable analog.

Cf. A269418, A269419.

The triangle without the zeros is A269919.

Sequence in context: A019750 A308473 A237289 * A247671 A011125 A197583

Adjacent sequences:  A238393 A238394 A238395 * A238397 A238398 A238399

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt, Feb 26 2014

STATUS

approved

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Last modified August 3 19:49 EDT 2020. Contains 336201 sequences. (Running on oeis4.)