cp's OEIS Frontend

A332648 Array read by antidiagonals: T(n,k) is the number of rooted unlabeled k-gonal cacti having n polygons.

%I A332648 #10 Dec 23 2020 01:51:23
%S A332648 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 A332648 46,37,48,1,1,1,4,22,62,208,111,115,1,1,1,5,25,140,333,1002,345,286,1,
%U A332648 1,1,5,37,176,985,1894,5012,1105,719,1,1,1,6,41,319,1397,7374,11258,25863,3624,1842,1
%N A332648 Array read by antidiagonals: T(n,k) is the number of rooted unlabeled k-gonal cacti having n polygons.
%C A332648 The number of nodes will be n*(k-1) + 1.
%H A332648 Andrew Howroyd, <a href="/A332648/b332648.txt">Table of n, a(n) for n = 0..1325</a>
%H A332648 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 A332648 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>
%H A332648 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%e A332648 Array begins:
%e A332648 ======================================================
%e A332648 n\k | 1   2    3     4     5      6      7       8
%e A332648 ----+-------------------------------------------------
%e A332648   0 | 1   1    1     1     1      1      1       1 ...
%e A332648   1 | 1   1    1     1     1      1      1       1 ...
%e A332648   2 | 1   2    2     3     3      4      4       5 ...
%e A332648   3 | 1   4    5    11    13     22     25      37 ...
%e A332648   4 | 1   9   13    46    62    140    176     319 ...
%e A332648   5 | 1  20   37   208   333    985   1397    3059 ...
%e A332648   6 | 1  48  111  1002  1894   7374  11757   31195 ...
%e A332648   7 | 1 115  345  5012 11258  57577 103376  331991 ...
%e A332648   8 | 1 286 1105 25863 68990 463670 937179 3643790 ...
%e A332648   ...
%o A332648 (PARI)
%o A332648 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A332648 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 A332648 T(n)={Mat(concat([vectorv(n+1,i,1)], vector(n,k,Col(R(n,k)))))}
%o A332648 { my(A=T(8)); for(n=1, #A, print(A[n,])) }
%Y A332648 Columns k=1..4 are A000012, A000081(n+1), A003080, A287891.
%Y A332648 Cf. A303694, A332649.
%K A332648 nonn,tabl
%O A332648 0,9
%A A332648 _Andrew Howroyd_, Feb 18 2020