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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.
13

%I #13 Feb 03 2021 21:48:56

%S 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 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 1,5,26,158,799,2597,4304,2171,235,1,1,1,6,32,245,1539,7286,20386,26492,9368,551

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

%C See section 4 and table 1 in the Labelle reference.

%H Andrew Howroyd, <a href="/A340811/b340811.txt">Table of n, a(n) for n = 0..1325</a>

%H 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 Array begins:

%e =======================================================

%e n\k | 2 3 4 5 6 7 8 9

%e ----+--------------------------------------------------

%e 0 | 1 1 1 1 1 1 1 1 ...

%e 1 | 1 1 1 1 1 1 1 1 ...

%e 2 | 1 1 1 1 1 1 1 1 ...

%e 3 | 2 2 3 3 4 4 5 5 ...

%e 4 | 3 5 8 11 16 20 26 32 ...

%e 5 | 6 12 32 56 103 158 245 343 ...

%e 6 | 11 39 141 359 799 1539 2737 4505 ...

%e 7 | 23 136 749 2597 7286 16970 35291 66603 ...

%e 8 | 47 529 4304 20386 71094 199879 483819 1045335 ...

%e ...

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o B(n,k)={my(p=1+O(x)); for(n=1, n, p=1+x*Ser(EulerT(Vec(p^(k-1))))); p}

%o C(p,k)={p(1) - x*p(1)^k + x*sumdiv(k, d, eulerphi(d)*p(d)^(k/d))/k}

%o 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 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 { Mat(vector(7, k, U(7, k+1)~)) }

%Y Columns 2..12 are A000055, A054581, A094610, A094611, A094637, A094651, A094652, A094653, A094654, A094655, A094656.

%Y Cf. A340812 (with oriented polygons).

%K nonn,tabl

%O 0,10

%A _Andrew Howroyd_, Feb 02 2021