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A332649 Array read by antidiagonals: T(n,k) is the number of unlabeled k-gonal cacti having n polygons.

%I A332649 #10 Dec 23 2020 01:51:28
%S A332649 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,3,1,1,1,1,3,4,6,1,1,1,1,3,7,8,
%T A332649 11,1,1,1,1,4,8,25,19,23,1,1,1,1,4,13,31,88,48,47,1,1,1,1,5,14,67,132,
%U A332649 366,126,106,1,1,1,1,5,20,80,372,636,1583,355,235,1
%N A332649 Array read by antidiagonals: T(n,k) is the number of unlabeled k-gonal cacti having n polygons.
%C A332649 The number of nodes will be n*(k-1) + 1.
%H A332649 Andrew Howroyd, <a href="/A332649/b332649.txt">Table of n, a(n) for n = 0..1325</a>
%H A332649 Maryam Bahrani and Jérémie Lumbroso, <a href="http://arxiv.org/abs/1608.01465">Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition</a>, arXiv:1608.01465 [math.CO], 2016.
%H A332649 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>
%H A332649 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%e A332649 Array begins:
%e A332649 ======================================================
%e A332649 n\k | 1  2   3    4    5     6     7     8      9
%e A332649 ----+-------------------------------------------------
%e A332649   0 | 1  1   1    1    1     1     1     1      1 ...
%e A332649   1 | 1  1   1    1    1     1     1     1      1 ...
%e A332649   2 | 1  1   1    1    1     1     1     1      1 ...
%e A332649   3 | 1  2   2    3    3     4     4     5      5 ...
%e A332649   4 | 1  3   4    7    8    13    14    20     22 ...
%e A332649   5 | 1  6   8   25   31    67    80   143    165 ...
%e A332649   6 | 1 11  19   88  132   372   504  1093   1391 ...
%e A332649   7 | 1 23  48  366  636  2419  3659  9722  13485 ...
%e A332649   8 | 1 47 126 1583 3280 16551 28254 91391 138728 ...
%e A332649 ...
%o A332649 (PARI) \\ here R(n,k) is column k+1 of A332648.
%o A332649 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A332649 R(n,k)={my(v=[]); for(n=1, n, my(g=1+x*Ser(v)); v=EulerT(Vec((g^k + g^(k%2)*subst(g^(k\2), x, x^2))/2))); concat([1], v)}
%o A332649 U(n,k)={my(p=Ser(R(n,k-1))); my(g(d)=subst(p + O(x*x^(n\d)), x, x^d)); Vec(g(1) + x*sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/(2*k) - x*(g(1)^k)/2 + x*if(k%2==0, g(2)^(k/2) - g(1)^2*g(2)^(k/2-1))/4)}
%o A332649 T(n)={Mat(concat([vectorv(n+1,i,1)], vector(n, k, Col(U(n,k+1)))))}
%o A332649 { my(A=T(8)); for(n=1, #A, print(A[n,])) }
%Y A332649 Columns k=1..4 are A000012, A000055(n+1), A003081, A287892.
%Y A332649 Cf. A303694, A332648, A332650, A332651.
%K A332649 nonn,tabl
%O A332649 0,14
%A A332649 _Andrew Howroyd_, Feb 18 2020