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A340811 Array read by antidiagonals: T(n,k) is the number of unlabeled k-gonal 2-trees with n polygons, n >= 0, k >= 2.

%I A340811 #13 Feb 03 2021 21:48:56
%S A340811 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,
%T A340811 32,39,23,1,1,1,4,16,56,141,136,47,1,1,1,5,20,103,359,749,529,106,1,1,
%U A340811 1,5,26,158,799,2597,4304,2171,235,1,1,1,6,32,245,1539,7286,20386,26492,9368,551
%N A340811 Array read by antidiagonals: T(n,k) is the number of unlabeled k-gonal 2-trees with n polygons, n >= 0, k >= 2.
%C A340811 See section 4 and table 1 in the Labelle reference.
%H A340811 Andrew Howroyd, <a href="/A340811/b340811.txt">Table of n, a(n) for n = 0..1325</a>
%H A340811 G. Labelle, C. Lamathe and P. Leroux, <a href="http://arXiv.org/abs/math.CO/0312424">Labeled and unlabeled enumeration of k-gonal 2-trees</a>, arXiv:math/0312424 [math.CO], Dec 23 2003.
%e A340811 Array begins:
%e A340811 =======================================================
%e A340811 n\k |  2   3    4     5     6      7      8       9
%e A340811 ----+--------------------------------------------------
%e A340811   0 |  1   1    1     1     1      1      1       1 ...
%e A340811   1 |  1   1    1     1     1      1      1       1 ...
%e A340811   2 |  1   1    1     1     1      1      1       1 ...
%e A340811   3 |  2   2    3     3     4      4      5       5 ...
%e A340811   4 |  3   5    8    11    16     20     26      32 ...
%e A340811   5 |  6  12   32    56   103    158    245     343 ...
%e A340811   6 | 11  39  141   359   799   1539   2737    4505 ...
%e A340811   7 | 23 136  749  2597  7286  16970  35291   66603 ...
%e A340811   8 | 47 529 4304 20386 71094 199879 483819 1045335 ...
%e A340811   ...
%o A340811 (PARI)
%o A340811 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A340811 B(n,k)={my(p=1+O(x)); for(n=1, n, p=1+x*Ser(EulerT(Vec(p^(k-1))))); p}
%o A340811 C(p,k)={p(1) - x*p(1)^k + x*sumdiv(k, d, eulerphi(d)*p(d)^(k/d))/k}
%o A340811 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)}
%o A340811 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}
%o A340811 { Mat(vector(7, k, U(7, k+1)~)) }
%Y A340811 Columns 2..12 are A000055, A054581, A094610, A094611, A094637, A094651, A094652, A094653, A094654, A094655, A094656.
%Y A340811 Cf. A340812 (with oriented polygons).
%K A340811 nonn,tabl
%O A340811 0,10
%A A340811 _Andrew Howroyd_, Feb 02 2021