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A342985
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Triangle read by rows: T(n,k) is the number of tree-rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.
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8
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1, 0, 0, 0, 2, 0, 0, 3, 3, 0, 0, 4, 36, 4, 0, 0, 5, 135, 135, 5, 0, 0, 6, 360, 1368, 360, 6, 0, 0, 7, 798, 7350, 7350, 798, 7, 0, 0, 8, 1568, 28400, 73700, 28400, 1568, 8, 0, 0, 9, 2826, 89073, 474588, 474588, 89073, 2826, 9, 0, 0, 10, 4770, 241220, 2292790, 4818092, 2292790, 241220, 4770, 10, 0
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OFFSET
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0,5
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COMMENTS
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The number of vertices is n + 2 - k.
For k >= 2, column k without the initial zero term is a polynomial of degree 4*(k-2)+1.
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LINKS
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FORMULA
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T(n,n+2-k) = T(n,k).
G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2),y) where G(x,y) + x*(1+y) is the g.f. of A342984.
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EXAMPLE
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Triangle begins:
1;
0, 0;
0, 2, 0;
0, 3, 3, 0;
0, 4, 36, 4, 0;
0, 5, 135, 135, 5, 0;
0, 6, 360, 1368, 360, 6, 0;
0, 7, 798, 7350, 7350, 798, 7, 0;
0, 8, 1568, 28400, 73700, 28400, 1568, 8, 0;
...
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PROG
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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*(1+y), v=Vec(sqrt(serreverse(x/g^2)/x))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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