A332648 Array read by antidiagonals: T(n,k) is the number of rooted unlabeled k-gonal cacti having n polygons.
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, 46, 37, 48, 1, 1, 1, 4, 22, 62, 208, 111, 115, 1, 1, 1, 5, 25, 140, 333, 1002, 345, 286, 1, 1, 1, 5, 37, 176, 985, 1894, 5012, 1105, 719, 1, 1, 1, 6, 41, 319, 1397, 7374, 11258, 25863, 3624, 1842, 1
Offset: 0
Examples
Array begins: ====================================================== n\k | 1 2 3 4 5 6 7 8 ----+------------------------------------------------- 0 | 1 1 1 1 1 1 1 1 ... 1 | 1 1 1 1 1 1 1 1 ... 2 | 1 2 2 3 3 4 4 5 ... 3 | 1 4 5 11 13 22 25 37 ... 4 | 1 9 13 46 62 140 176 319 ... 5 | 1 20 37 208 333 985 1397 3059 ... 6 | 1 48 111 1002 1894 7374 11757 31195 ... 7 | 1 115 345 5012 11258 57577 103376 331991 ... 8 | 1 286 1105 25863 68990 463670 937179 3643790 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325
- Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
- Wikipedia, Cactus graph
- Index entries for sequences related to cacti
Programs
-
PARI
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} 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)} T(n)={Mat(concat([vectorv(n+1,i,1)], vector(n,k,Col(R(n,k)))))} { my(A=T(8)); for(n=1, #A, print(A[n,])) }
Comments