A342987 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.
1, 0, 1, 0, 2, 2, 0, 3, 15, 5, 0, 4, 60, 84, 14, 0, 5, 175, 650, 420, 42, 0, 6, 420, 3324, 5352, 1980, 132, 0, 7, 882, 13020, 42469, 37681, 9009, 429, 0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 1430, 0, 9, 2970, 118998, 1142622, 3462354, 3711027, 1421226, 175032, 4862
Offset: 0
Examples
Triangle begins: 1; 0, 1; 0, 2, 2; 0, 3, 15, 5; 0, 4, 60, 84, 14; 0, 5, 175, 650, 420, 42; 0, 6, 420, 3324, 5352, 1980, 132; 0, 7, 882, 13020, 42469, 37681, 9009, 429; 0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 1430; ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
- T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VIIIb.
Crossrefs
Programs
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PARI
\\ here G(n,y) is A342984 as g.f. F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)} G(n,y)={my(g=F(n,y)); subst(g, x, serreverse(x*g^2))} H(n)={my(g=G(n,y)-x, v=Vec(sqrt(serreverse(x/g^2)/x))); [Vecrev(t) | t<-v]} { my(T=H(8)); for(n=1, #T, print(T[n])) }
Formula
G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2,y) where G(x,y) + x is the g.f. of A342984.
Comments