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%I #19 Apr 15 2021 07:40:40
%S 1,0,1,0,1,2,0,1,7,5,0,1,16,37,14,0,1,30,150,176,42,0,1,50,449,1104,
%T 794,132,0,1,77,1113,4795,7077,3473,429,0,1,112,2422,16456,41850,
%U 41504,14893,1430,0,1,156,4788,47832,189183,319320,228810,63004,4862
%N Triangle read by rows: T(n,k) is the number of rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.
%C The number of vertices is n + 2 - k.
%C For k >= 2, column k is a polynomial of degree 3*(k-2). This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.
%C By duality, also the number of loopless rooted planar maps with n edges and k vertices.
%H Andrew Howroyd, <a href="/A342981/b342981.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus. III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VIb.
%F G.f. A(x,y) satisfies A(x) = G(x*A(x,y)^2, y) where G(x,y) = 1 + x*y + x*B(x,y) and B(x,y) is the g.f. of A082680.
%F A027836(n+1) = Sum_{k=1..n+1} k*T(n,k).
%F A002293(n) = Sum_{k=1..n+1} k*T(n,n+2-k).
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 1, 7, 5;
%e 0, 1, 16, 37, 14;
%e 0, 1, 30, 150, 176, 42;
%e 0, 1, 50, 449, 1104, 794, 132;
%e 0, 1, 77, 1113, 4795, 7077, 3473, 429;
%e 0, 1, 112, 2422, 16456, 41850, 41504, 14893, 1430;
%e ...
%t G[m_, y_] := Sum[x^n*Sum[(n + k - 1)!*(2*n - k)!*y^k/(k!*(n + 1 - k)!*(2*k - 1)!*(2*n - 2*k + 1)!), {k, 1, n}], {n, 1, m}] + O[x]^m;
%t H[n_] := With[{g = 1 + x*y + x*G[n - 1, y]}, Sqrt[InverseSeries[x/g^2 + O[x]^(n + 1), x]/x]];
%t CoefficientList[#, y]& /@ CoefficientList[H[10], x] // Flatten (* _Jean-François Alcover_, Apr 15 2021, after _Andrew Howroyd_ *)
%o (PARI) \\ here G(n, y) gives A082680 as g.f.
%o G(n,y)={sum(n=1, n, x^n*sum(k=1, n, (n+k-1)!*(2*n-k)!*y^k/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!))) + O(x*x^n)}
%o H(n)={my(g=1+x*y+x*G(n-1, y), v=Vec(sqrt(serreverse(x/g^2)/x))); vector(#v, n, Vecrev(v[n], n))}
%o { my(T=H(8)); for(n=1, #T, print(T[n])) }
%Y Columns k=3..4 are A005581, A006468.
%Y Diagonals are A000108, A006419, A006420, A006421.
%Y Row sums are A000260.
%Y Cf. A002293, A027836, A082680, A269920, A342980, A343092.
%K nonn,tabl
%O 0,6
%A _Andrew Howroyd_, Apr 02 2021
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