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A342989
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Triangle read by rows: T(n,k) is the number of nonseparable rooted toroidal maps with n edges and k faces, n >= 2, k = 1..n-1.
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7
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1, 4, 4, 10, 39, 10, 20, 190, 190, 20, 35, 651, 1568, 651, 35, 56, 1792, 8344, 8344, 1792, 56, 84, 4242, 33580, 64667, 33580, 4242, 84, 120, 8988, 111100, 361884, 361884, 111100, 8988, 120, 165, 17490, 317680, 1607125, 2713561, 1607125, 317680, 17490, 165
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OFFSET
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2,2
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COMMENTS
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The number of vertices is n - k.
Column k is a polynomial of degree 3*k. This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.
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LINKS
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FORMULA
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T(n,n-k) = T(n,k).
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EXAMPLE
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Triangle begins:
1;
4, 4;
10, 39, 10;
20, 190, 190, 20;
35, 651, 1568, 651, 35;
56, 1792, 8344, 8344, 1792, 56;
84, 4242, 33580, 64667, 33580, 4242, 84;
120, 8988, 111100, 361884, 361884, 111100, 8988, 120;
...
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PROG
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(PARI)
MQ(n, g, x=1)={ \\ after Quadric in A269921.
my(Q=matrix(n+1, g+1)); Q[1, 1]=x;
for(n=1, n, for(g=0, min(n\2, g),
my(t = (1+x)*(2*n-1)/3 * Q[n, 1+g]
+ if(g && n>1, (2*n-3)*(2*n-2)*(2*n-1)/12 * Q[n-1, g])
+ sum(k = 1, n-1, sum(i = 0, g, (2*k-1) * (2*(n-k)-1) * Q[k, 1+i] * Q[n-k, 1+g-i]))/2);
Q[1+n, 1+g] = t * 6/(n+1); ));
Q
}
F(n, m, y, z)={my(Q=MQ(n, m, z)); sum(n=0, n, x^n*Ser(Q[1+n, ]/z, y)) + O(x*x^n)}
H(n, g=1)={my(p=F(n, g, 'y, 'z), v=Vec(polcoef(subst(p, x, serreverse(x*p^2)), g, 'y))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(10)); 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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