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A370948 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.

%I A370948 #17 Mar 28 2024 23:03:37
%S A370948 1,3,1,22,15,1,262,271,53,1,4336,6020,2085,165,1,91984,160336,81310,
%T A370948 13040,487,1,2381408,4996572,3364011,851690,73024,1407,1,72800928,
%U A370948 178613156,150499951,53119521,7696794,383649,4041,1
%N A370948 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 A370948 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 A370948 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 A370948 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 A370948 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 A370948 T(n,0) = A005264(n).
%F A370948 T(n,n-1) = 1.
%e A370948 Triangle T(n,k) begins:
%e A370948 n\k    0     1     2     3     4 ...
%e A370948 1      1;
%e A370948 2      3,    1;
%e A370948 3     22,   15,    1;
%e A370948 4    262,  271,   53,    1;
%e A370948 5   4336, 6020, 2085,  165,    1;
%e A370948 ...
%o A370948 (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 A370948 { my(A=T(8)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Mar 06 2024
%Y A370948 Cf. A364709, A005264 (k=0), A370949.
%Y A370948 Row sums are A364816.
%Y A370948 Series reversion as e.g.f. is related to A092271.
%K A370948 nonn,tabl
%O A370948 1,2
%A A370948 _Paul Laubie_, Mar 06 2024