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