OFFSET
0,10
COMMENTS
See section 3 of the Labelle reference.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.
FORMULA
G.f. of column k: B(x) - x*B(x)^k + x*(Sum_{d|k} phi(d)*B(x^d)^(k/d))/k, where B(x) if the g.f. of column k of A340814.
EXAMPLE
Array begins:
=========================================================
n\k | 2 3 4 5 6 7 8 9
----+----------------------------------------------------
0 | 1 1 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 1 1 ...
2 | 1 1 1 1 1 1 1 1 ...
3 | 2 2 3 3 4 4 5 5 ...
4 | 3 7 11 17 25 33 43 55 ...
5 | 6 18 49 96 177 285 442 635 ...
6 | 11 68 252 687 1537 3014 5370 8901 ...
7 | 23 251 1406 5087 14310 33632 70000 132533 ...
8 | 47 1020 8405 40546 141582 399065 966254 2089103 ...
...
PROG
(PARI) \\ here B(n, k) gives column k of A340814.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(n, k)={my(p=1+O(x)); for(n=1, n, p=1+x*Ser(EulerT(Vec(p^(k-1))))); p}
C(n, k)={my(p=B(n, k)); Vec(p - x*p^k + x*sumdiv(k, d, eulerphi(d)*subst(p + O(x*x^(n\d)), x, x^d)^(k/d))/k)}
{ Mat(vector(7, k, C(7, k+1)~)) }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 02 2021
STATUS
approved