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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-François 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.)