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%I #17 Mar 28 2024 23:03:37
%S 1,3,1,22,15,1,262,271,53,1,4336,6020,2085,165,1,91984,160336,81310,
%T 13040,487,1,2381408,4996572,3364011,851690,73024,1407,1,72800928,
%U 178613156,150499951,53119521,7696794,383649,4041,1
%N Triangle read by rows: T(n,k) is the number of labeled forests of rooted Greg hypertrees with n white vertices and weight k, 0 <= k < n.
%C 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 The weight of a forest of rooted Greg hypertrees is the number of hypertrees minus 1 plus the weight of each hyperedge which is the number of vertices it connects minus 2. See A364709 for the analog sequence for hypertrees. A forest of rooted Greg hypertrees of weight 0 is exactly a Greg tree.
%H 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 E.g.f: series reversion in t of (log(1+v*t)/v - exp(t) + t + 1)*exp(-t), where the formal variable v encodes the weight.
%F T(n,0) = A005264(n).
%F T(n,n-1) = 1.
%e Triangle T(n,k) begins:
%e n\k 0 1 2 3 4 ...
%e 1 1;
%e 2 3, 1;
%e 3 22, 15, 1;
%e 4 262, 271, 53, 1;
%e 5 4336, 6020, 2085, 165, 1;
%e ...
%o (PARI) T(n)={my(x='x+O('x^(n+1))); [Vecrev(p) | p <- Vec(serlaplace(serreverse( (log(1+y*x)/y - exp(x) + x + 1)*exp(-x) )))]}
%o { my(A=T(8)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Mar 06 2024
%Y Cf. A364709, A005264 (k=0), A370949.
%Y Row sums are A364816.
%Y Series reversion as e.g.f. is related to A092271.
%K nonn,tabl
%O 1,2
%A _Paul Laubie_, Mar 06 2024
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