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A069728
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Number of rooted non-separable Eulerian planar maps with n edges.
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2
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1, 1, 1, 1, 2, 6, 19, 64, 230, 865, 3364, 13443, 54938, 228749, 967628, 4149024, 18000758, 78905518, 349037335, 1556494270, 6991433386, 31609302688, 143755711433, 657301771172, 3020175361634, 13939605844996, 64604720622719
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OFFSET
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0,5
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LINKS
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FORMULA
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G.f.: y = A(x) satisfies 0 = y^5 - y^4 - 12*x*y^3 + x*(16*x + 11)*y^2 - 8*x^2*y + x^2. - Gheorghe Coserea, Apr 12 2018
A(x) = 1 + serreverse((1+x)^2*(1+12*x-(1-4*x)^(3/2))/(2*(4*x+3)^2)); equivalently, it can be rewritten as A(x) = 1 + serreverse((y-1)*(y^2+y-1)^2/(y^3*(3*y-2)^2)), where y = A000108(x). - Gheorghe Coserea, Apr 14 2018
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EXAMPLE
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A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 64*x^7 + 230*x^8 + ...
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MATHEMATICA
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Flatten[{1, Table[(Sum[(-1)^j*Binomial[2*n + j - 1, j] * Sum[(-1)^k*2^(n - j - k - 1)*Binomial[j, k] * Binomial[2*n, n - j - k - 1], {k, 0, Min[j, n - j - 1]}], {j, 0, n - 1}] - 2*Sum[(-1)^j*Binomial[2*n + j - 1, j] * Sum[(-1)^k*2^(n - j - k - 2) * Binomial[j, k]*Binomial[2*n, n - j - k - 2], {k, 0, Min[j, n - j - 2]}], {j, 0, n - 2}])/n, {n, 1, 30}]}] (* Vaclav Kotesovec, Apr 13 2018 *) (* In the article by Liskovets and Walsh, p. 218, E'ns(n), the factor -2*Sum[...] is missing. *)
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PROG
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(PARI)
seq(N) = {
my(x ='x+O('x^N), y=serreverse(x*(1+x/2-x^2/4)^2/(2*(1+x)^2)));
Vec(1+y/2-y^2/4);
};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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