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Array read by antidiagonals: T(n,k) is the number of rooted unlabeled k-gonal cacti having n polygons.
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%I #10 Dec 23 2020 01:51:23

%S 1,1,1,1,1,1,1,1,2,1,1,1,2,4,1,1,1,3,5,9,1,1,1,3,11,13,20,1,1,1,4,13,

%T 46,37,48,1,1,1,4,22,62,208,111,115,1,1,1,5,25,140,333,1002,345,286,1,

%U 1,1,5,37,176,985,1894,5012,1105,719,1,1,1,6,41,319,1397,7374,11258,25863,3624,1842,1

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

%C The number of nodes will be n*(k-1) + 1.

%H Andrew Howroyd, <a href="/A332648/b332648.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

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

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

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

%e 2 | 1 2 2 3 3 4 4 5 ...

%e 3 | 1 4 5 11 13 22 25 37 ...

%e 4 | 1 9 13 46 62 140 176 319 ...

%e 5 | 1 20 37 208 333 985 1397 3059 ...

%e 6 | 1 48 111 1002 1894 7374 11757 31195 ...

%e 7 | 1 115 345 5012 11258 57577 103376 331991 ...

%e 8 | 1 286 1105 25863 68990 463670 937179 3643790 ...

%e ...

%o (PARI)

%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 T(n)={Mat(concat([vectorv(n+1,i,1)], vector(n,k,Col(R(n,k)))))}

%o { my(A=T(8)); for(n=1, #A, print(A[n,])) }

%Y Columns k=1..4 are A000012, A000081(n+1), A003080, A287891.

%Y Cf. A303694, A332649.

%K nonn,tabl

%O 0,9

%A _Andrew Howroyd_, Feb 18 2020