The On-Line Encyclopedia of Integer Sequences (OEIS)

Revisions by Paul Laubie (See also Paul Laubie's wiki page)

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A370360

Number of labeled semisimple rings with n elements.
(history; published version)

#50 by Paul Laubie at Thu Mar 28 20:18:16 EDT 2024

STATUS

editing

proposed

#49 by Paul Laubie at Thu Mar 28 20:17:53 EDT 2024

LINKS

P. Laubie, <a href="https://github.com/Kellaubz/labeled_semisimple_rings">A Github repository with a code to compute the terms of the form a(p^n)</a>

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.
(history; published version)

#4 by Paul Laubie at Wed Mar 06 12:03:01 EST 2024

STATUS

editing

proposed

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.
(history; published version)

#9 by Paul Laubie at Wed Mar 06 12:02:39 EST 2024

STATUS

editing

proposed

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.
(history; published version)

#3 by Paul Laubie at Wed Mar 06 12:00:53 EST 2024

NAME

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.

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.
(history; published version)

#8 by Paul Laubie at Wed Mar 06 12:00:12 EST 2024

NAME

Number 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.

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.
(history; published version)

#2 by Paul Laubie at Wed Mar 06 11:58:58 EST 2024

NAME

allocated for Paul LaubieTriangle read by rows: T(n,k) is the number of forests of labeled rooted Greg hypertrees with n vertices and k black vertices, 0 <= k < n.

DATA

1, 3, 1, 19, 16, 3, 189, 268, 115, 15, 2576, 5221, 3655, 1050, 105, 44683, 118599, 117236, 54040, 11655, 945, 941977, 3102184, 3996384, 2581138, 883575, 152460, 10395, 23388025, 92149019, 147043422, 123318510, 58806055, 15980580, 2297295, 135135

OFFSET

1,2

COMMENTS

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 have at least two children.

See A048160 for the analog sequence for Greg trees.

LINKS

Paul Laubie, <a href="https://arxiv.org/abs/2401.17439">Hypertrees and embedding of the FMan operad</a>, arXiv:2401.17439 [math.QA], 2024.

FORMULA

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.

a(n,0) = A052888(n).

a(n,n-1) = A001147(n).

EXAMPLE

Triangle T(n,k) begins:

n\k 0 1 2 3 4...

1 1;

2 3, 1;

3 19, 16, 3;

4 189, 268, 115, 15;

5 2576, 5221, 3655, 1050, 105;

...

CROSSREFS

Cf. A048160, A052888 (k=0), A001147 (k=n-1).

Row sums are A364816.

KEYWORD

allocated

nonn,tabl

AUTHOR

Paul Laubie, Mar 06 2024

STATUS

approved

editing

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.
(history; published version)

#7 by Paul Laubie at Wed Mar 06 11:40:32 EST 2024

EXAMPLE

n\k 0 1 2 3 4 ...

#6 by Paul Laubie at Wed Mar 06 11:36:06 EST 2024

FORMULA

a(n,kn-1) = 1.

#5 by Paul Laubie at Wed Mar 06 11:30:02 EST 2024

CROSSREFS

Cf. A364709, A005264 (k=0), A370949.