A005264 -id:A005264 - cp's OEIS frontend

A364816 Number of labeled forests of rooted Greg hypertrees with n white vertices.

1, 4, 38, 587, 12607, 347158, 11668113, 463118041, 21199488803, 1099465138203, 63715991036964, 4080500855334901, 286178278238641752, 21813909692571410084, 1795659553423061982001, 158754024731440581761116, 15002712207593790179795284, 1509215071938528737864389367, 161017605699030302902310357883

Offset: 1

Paul Laubie, Oct 21 2023

A Greg hypertree is a hypertree with black and white vertices, such that black vertices are unlabeled and have at least two incoming edges.

Cf. A005264, A052888.

Rest[CoefficientList[InverseSeries[Series[E^-x (1 + x + Log[1 + x]) - 1, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Oct 24 2023 *)

my(t='t+O('t^25)); Vec(serlaplace(serreverse((log(1+t)-exp(t)+t+1)*exp(-t)))) \\ Michel Marcus, Oct 21 2023

E.g.f: series reversion of (log(1+t)-exp(t)+t+1)*exp(-t).

a(n) ~ sqrt((1+s)*(2+s)/((1+r)*(3 + s*(3+s)))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where s = 0.3900539630495916058133890253422601894372373496844... is the root of the equation exp(-s + 1/(1+s)) = 1+s and r = exp(-s)*(1 + 1/(1+s)) - 1 = 0.1640664235584946357534702598223332293549130374395... - Vaclav Kotesovec, Oct 24 2023

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.

Original entry on oeis.org

1, 3, 1, 22, 15, 1, 262, 271, 53, 1, 4336, 6020, 2085, 165, 1, 91984, 160336, 81310, 13040, 487, 1, 2381408, 4996572, 3364011, 851690, 73024, 1407, 1, 72800928, 178613156, 150499951, 53119521, 7696794, 383649, 4041, 1

Offset: 1

Paul Laubie, Mar 06 2024

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.

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.

			Triangle T(n,k) begins:
n\k    0     1     2     3     4 ...
1      1;
2      3,    1;
3     22,   15,    1;
4    262,  271,   53,    1;
5   4336, 6020, 2085,  165,    1;
...

Paul Laubie, Hypertrees and embedding of the FMan operad, arXiv:2401.17439 [math.QA], 2024.

Cf. A364709, A005264 (k=0), A370949.
Row sums are A364816.
Series reversion as e.g.f. is related to A092271.

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) )))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Mar 06 2024

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.
T(n,0) = A005264(n).
T(n,n-1) = 1.

cp's OEIS Frontend

A364816 Number of labeled forests of rooted Greg hypertrees with n white vertices.

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