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A340811
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Array read by antidiagonals: T(n,k) is the number of unlabeled k-gonal 2-trees with n polygons, n >= 0, k >= 2.
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13
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1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 3, 5, 6, 1, 1, 1, 3, 8, 12, 11, 1, 1, 1, 4, 11, 32, 39, 23, 1, 1, 1, 4, 16, 56, 141, 136, 47, 1, 1, 1, 5, 20, 103, 359, 749, 529, 106, 1, 1, 1, 5, 26, 158, 799, 2597, 4304, 2171, 235, 1, 1, 1, 6, 32, 245, 1539, 7286, 20386, 26492, 9368, 551
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OFFSET
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0,10
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COMMENTS
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See section 4 and table 1 in the Labelle reference.
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LINKS
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EXAMPLE
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Array begins:
=======================================================
n\k | 2 3 4 5 6 7 8 9
----+--------------------------------------------------
0 | 1 1 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 1 1 ...
2 | 1 1 1 1 1 1 1 1 ...
3 | 2 2 3 3 4 4 5 5 ...
4 | 3 5 8 11 16 20 26 32 ...
5 | 6 12 32 56 103 158 245 343 ...
6 | 11 39 141 359 799 1539 2737 4505 ...
7 | 23 136 749 2597 7286 16970 35291 66603 ...
8 | 47 529 4304 20386 71094 199879 483819 1045335 ...
...
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PROG
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(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(n, k)={my(p=1+O(x)); for(n=1, n, p=1+x*Ser(EulerT(Vec(p^(k-1))))); p}
C(p, k)={p(1) - x*p(1)^k + x*sumdiv(k, d, eulerphi(d)*p(d)^(k/d))/k}
S(p, k)={my(p2=p(2)); if(k%2, 1+x*Ser(EulerT(Vec(x*p2^(k\2) + x^2*(p2^(k-1) - p(4)^(k\2))/2 ))), my(r=p2^(k/2-1), q=1+O(x)); while(serprec(q, x)<serprec(p2, x), my(t=r*q); q=1+x*Ser(EulerT(Vec(x*t + x^2*subst(p(1)^(k-1) - t, x, x^2)/2)))); q + x*p2^(k/2-1)*(p2-q^2)/2)}
U(n, k)={my(b=B(n, k), p(d)=subst(b + O(x*x^(n\d)), x, x^d)); Vec(C(p, k) + S(p, k))/2}
{ Mat(vector(7, k, U(7, k+1)~)) }
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CROSSREFS
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Columns 2..12 are A000055, A054581, A094610, A094611, A094637, A094651, A094652, A094653, A094654, A094655, A094656.
Cf. A340812 (with oriented polygons).
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KEYWORD
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AUTHOR
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STATUS
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approved
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