A006301 Number of rooted genus-2 maps with n edges. (Formerly M5120)

0, 0, 0, 0, 21, 966, 27954, 650076, 13271982, 248371380, 4366441128, 73231116024, 1183803697278, 18579191525700, 284601154513452, 4272100949982600, 63034617139799916, 916440476048146056, 13154166812674577412, 186700695099591735024, 2623742783421329300190, 36548087103760045010148, 505099724454854883618924

Offset

0,5

References

E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Links

T. D. Noe, Table of n, a(n) for n=0..30 (from Mednykh and Nedela)

E. A. Bender and E. R. Canfield, The number of rooted maps on an orientable surface, J. Combin. Theory, B 53 (1991), 293-299.

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

S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.

Mathematica

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, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 20 2018 *)

Prog

(PARI)
A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A006301_ser(N) = {
  my(y=A005159_ser(N+1));
  -y*(y-1)^4*(4*y^4 - 16*y^3 + 153*y^2 - 148*y + 196)/(9*(y-2)^7*(y+2)^4);
};
concat([0, 0, 0, 0], Vec(A006301_ser(19))) \\ Gheorghe Coserea, Jun 02 2017

Crossrefs

Column k=2 of A238396.

Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, this sequence, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

Sequence in context: A015305 A381745 A101732 * A220384 A184133 A004704

Adjacent sequences: A006298 A006299 A006300 * A006302 A006303 A006304

Keyword

nonn,changed

Author

N. J. A. Sloane and Simon Plouffe

Extensions

More terms from Joerg Arndt, Feb 26 2014

Status

approved