A380241 - OEIS

%I #14 Jan 22 2025 17:33:01

%S 1,1,1,1,1,1,1,2,1,1,1,5,9,1,1,1,14,100,54,1,1,1,42,1225,3000,378,1,1,

%T 1,132,15876,171500,110000,2916,1,1,1,429,213444,10001880,30012500,

%U 4550000,24057,1,1,1,1430,2944656,591666768,7981500240,5987493750,204000000,208494,1,1

%N Array read by antidiagonals: T(n,k) is the number of rooted (2k)-regular planar maps with n vertices, n >= 0, k >= 0.

%C The zeroth column is included by convention only for consistency with the first row sequences.

%C The case for regular planar maps of odd valency is more complicated and without simple closed form formulas, so not presented in this sequence. See the references for additional information.

%H Andrew Howroyd, <a href="/A380241/b380241.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)

%H E. A. Bender and E. R. Canfield, <a href="https://doi.org/10.1137/S089548019017765">The number of degree restricted rooted maps on the sphere</a>, SIAM J. Discrete Math. 7 (1994) 9-15.

%H Zhicheng Gao and Mizan Rahman, <a href="https://doi.org/10.1007/BF02558463">Enumeration of k-poles</a>, Annals of Combinatorics 1 (1997), pp. 55-66.

%H W. T. Tutte, <a href="https://doi.org/10.4153/CJM-1962-061-1">A Census of Slicings</a>, Canad. J. Math. 14 (1962), 708-722.

%H W. T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1963-029-x">A Census of Planar Maps</a>, Canad. J. Math. 15 (1963), 249-271.

%F T(n,k) =  2*binomial(2*k-1, k)^n*(n*k)!/(n!*(n*k - n + 2)!) for k > 0.

%e Array begins:

%e ====================================================================

%e n\k | 0  1      2          3               4                   5 ...

%e ----+---------------------------------------------------------------

%e   0 | 1  1      1          1               1                   1 ...

%e   1 | 1  1      2          5              14                  42 ...

%e   2 | 1  1      9        100            1225               15876 ...

%e   3 | 1  1     54       3000          171500            10001880 ...

%e   4 | 1  1    378     110000        30012500          7981500240 ...

%e   5 | 1  1   2916    4550000      5987493750       7304332956480 ...

%e   6 | 1  1  24057  204000000   1302227368750    7310748066293952 ...

%e   7 | 1  1 208494 9690000000 301107909375000 7794097754539041792 ...

%e   ...

%o (PARI) T(n,k)=if(k==0, 1, 2*binomial(2*k-1,k)^n*(n*k)!/(n!*(n*k - n + 2)!))

%Y Columns 0..3 are A000012 twice, A000168, A380242.

%Y Rows 0..3 are A000012, A000108, A060150, A380243.

%Y Cf. A269920.

%K nonn,tabl,new

%O 0,8

%A _Andrew Howroyd_, Jan 22 2025