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A370949 Triangle read by rows: T(n,k) is the number of forests of labeled rooted Greg hypertrees with n white vertices and k black vertices, 0 <= k < n.

%I A370949 #9 Mar 28 2024 23:37:51
%S A370949 1,3,1,19,16,3,189,268,115,15,2576,5221,3655,1050,105,44683,118599,
%T A370949 117236,54040,11655,945,941977,3102184,3996384,2581138,883575,152460,
%U A370949 10395,23388025,92149019,147043422,123318510,58806055,15980580,2297295,135135
%N A370949 Triangle read by rows: T(n,k) is the number of forests of labeled rooted Greg  hypertrees with n white vertices and k black vertices, 0 <= k < n.
%C A370949 A rooted Greg hypertree is a hypertree with black and white vertices such that white vertices are labeled, black vertices are unlabeled, and each black vertex has at least two children.
%C A370949 See A048160 for the analog sequence for Greg trees.
%H A370949 Paul Laubie, <a href="https://arxiv.org/abs/2401.17439">Hypertrees and embedding of the FMan operad</a>, arXiv:2401.17439 [math.QA], 2024.
%F A370949 E.g.f.: series reversion in t of (log(1+t) - u*exp(t) + u*t + u)*exp(-t), where the formal variable u encodes the number of black vertices.
%F A370949 T(n,0) = A052888(n).
%F A370949 T(n,n-1) = A001147(n).
%e A370949 Triangle T(n,k) begins:
%e A370949 n\k    0     1     2     3     4 ...
%e A370949 1      1;
%e A370949 2      3,    1;
%e A370949 3     19,   16,    3;
%e A370949 4    189,  268,  115,   15;
%e A370949 5   2576, 5221, 3655, 1050,  105;
%e A370949 ...
%o A370949 (PARI) T(n)={my(x='x+O('x^(n+1))); [Vecrev(p) | p <- Vec(serlaplace(serreverse( (log(1+x) - y*exp(x) + y*x + y)*exp(-x) )))]}
%o A370949 { my(A=T(8)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Mar 06 2024
%Y A370949 Cf. A048160, A052888 (k=0), A001147 (k=n-1).
%Y A370949 Row sums are A364816.
%K A370949 nonn,tabl
%O A370949 1,2
%A A370949 _Paul Laubie_, Mar 06 2024