Paul Laubie - cp's OEIS frontend

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.

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

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.

See A048160 for the analog sequence for Greg trees.

			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;
...

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

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

Row sums are A364816.

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) )))]}
{ 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+t) - u*exp(t) + u*t + u)*exp(-t), where the formal variable u encodes the number of black vertices.
T(n,0) = A052888(n).
T(n,n-1) = A001147(n).

A370360 Number of labeled semisimple rings with n elements.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 24409921536000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 1

Paul Laubie, Mar 05 2024

nonn

Using the Artin-Wedderburn theorem, a finite semisimple ring is a product of matrix algebras over finite field. In particular, if n is squarefree then any semisimple ring of cardinal n is commutative. One can be more precise, indeed all semisimple rings with n elements are commutative if and only if the only 4th power that divides n is 1.
The analogous sequences for abelian groups and cyclic groups are A034382 and A034381, respectively.
In the case of commutative semisimple rings, we get the factorial numbers.

			For n=4, we have two possible rings: F_4 and F_2 X F_2. We use the notation F_q to denote the finite field with q elements. To compute a(4) we need to know how many ring automorphisms F_4 and F_2 X F_2 admit. For F_4, we have that Aut(F_4) is generated by the Frobenius morphism, hence we have 2 automorphisms. For F_2 X F_2, the only nontrivial automorphism is exchanging the two coordinates, hence we also have 2 automorphisms. Hence:
a(4) = 24/2 + 24/2 = 24.
We can compute a(2^k) for some small values of k:
a(4) = 4! = 24,
a(8) = 8!,
a(16) = 16! + 16!/6,
a(32) = 32! + 32!/6,
a(64) = 64! + 64!/12 + 64!/12,
a(128) = 128! + 128!/36 + 128!/18 + 128!/12,
...

Paul Laubie, A Github repository with a code to compute the terms of the form a(p^n).

Cf. A000142, A005117, A038538.

Cf. A034382, A034381.

If n is squarefree then we have a(n) = n!. More precisely, a(n) = n! if and only if the only 4th power that divides n is 1. In particular, n=16 is the smallest n such that a(n) is different from n!.

If n and m are relatively prime, then a(n*m) = (n*m)!*a(n)*a(m)/(n!*m!).

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.

A370677 a(n) is the dimension of the multilinear part of the free flexible Lie-admissible algebra with n generators.

Original entry on oeis.org

1, 2, 8, 48, 384, 3841

Offset: 1

Paul Laubie, Feb 26 2024

The flexible identity is (x*y)*z + (z*y)*x = x*(y*z) + z*(y*x).
The Lie-admissible identity is (x*y)*z - x*(y*z) + z*(y*x) - (z*y)*x + y*(x*z) - (y*x)*z = (x*z)*y - x*(z*y) + y*(z*x) - (y*z)*x + z*(x*y) - (z*x)*y.
Flexible Lie-admissible algebras are algebras (possibly non-associative) satisfying those two identities.

Hyo Chul Myung, Flexible Lie-Admissible algebras (Thesis (Ph.D.)-Michigan State University), ProQuest LLC, Ann Arbor, MI, 1971.

A370161 a(n) is the dimension of the multilinear part of the free flexible algebra with n generators.

Original entry on oeis.org

1, 2, 9, 61, 545, 5986

Offset: 1

Paul Laubie, Feb 26 2024

The flexible identity is (x*y)*z + (z*y)*x = x*(y*z) + z*(y*x).

Flexible algebras are algebras (possibly non-associative) satisfying this identity.

Cf. A370677 for flexible Lie-admissible algebras.

A367753 Number of shapes of labeled forest of rooted hypertrees with n vertices.

Original entry on oeis.org

1, 2, 8, 55, 507, 5969, 85605, 1445420, 28110615, 618760615, 15207174501, 412790636977, 12265337498000, 395962288858946, 13800754780797740, 516494067220932259, 20658199248901273576, 879406216174705907137, 39698270229941320201019, 1894212537494300993244732

Offset: 1

Paul Laubie, Nov 29 2023

nonn

See A367752 for the definition of the shape of a labeled rooted hypertree.

Cf. A052888, A210586, A364709, A364816, A367752.

my(x='x+O('x^30)); Vec(serlaplace(serreverse((1+x)*log(1+log(1+x))*exp(-x)))) \\ Michel Marcus, Nov 30 2023

R.=PowerSeriesRing(QQ,30);(1+t)*(ln(1+ln(1+t))*exp(-t)).reverse().egf_to_ogf().list()[1:]

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

A367752 Number of shapes of labeled rooted hypertrees with n vertices.

Original entry on oeis.org

1, 1, 4, 29, 256, 3007, 42932, 721121, 13982563, 306967231, 7527903208, 203977383469, 6051630040496, 195111205542541, 6792697846367791, 253966747582533681, 10149075292428481965, 431705938073882999275, 19474660918369182445456, 928660364396786865580881

Offset: 1

Paul Laubie, Nov 29 2023

nonn

The shape of a labeled rooted hypertree is a labeled rooted hypertrees where we replace all the maximal subtrees by a corolla rooted on a new unlabeled black vertex.
If we remove the black vertices that are the parent of only 1 white vertex, we obtain labeled rooted hypertrees with black and white vertices such that:
  - black vertices are unlabeled;
  - black vertices have at least two children;
  - the children of a black vertex are white, and are connected to it via simple edges (edges connecting only two vertices);
  - the children of a white vertex are connected to it via hyperedges (edges connecting strictly more than two vertices).

			For n = 3 the a(3) = 4 solutions are:
  - the corolla with a black root which have 3 white children,
  - and the 3 possible labeling of the hypertree with a white root which have 2 white children connected to it via a hyperedge.

Cf. A052888, A210586, A364709, A364816.

my(x='x+O('x^30)); Vec(serlaplace(serreverse(log(1+x)*exp(-exp(x)+x+1)))) \\ Michel Marcus, Nov 30 2023

R.=PowerSeriesRing(QQ);(ln(1+t)*exp(-exp(t)+t+1)).reverse().egf_to_ogf().list()[1:]

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

A364709 Triangle read by rows: T(n,k) is the number of forests of labeled rooted hypertrees with n vertices and weight k, 0 <= k < n.

Original entry on oeis.org

1, 2, 1, 9, 9, 1, 64, 96, 28, 1, 625, 1250, 625, 75, 1, 7776, 19440, 14040, 3240, 186, 1, 117649, 352947, 336140, 120050, 14749, 441, 1, 2097152, 7340032, 8716288, 4300800, 870912, 61824, 1016, 1, 43046721, 172186884, 245525742, 156243654, 45605511, 5664330, 245025, 2295, 1

Offset: 1

Paul Laubie, Oct 20 2023

The weight is the number of hypertrees minus 1 plus the weight of each hyperedge which is the number of vertices it connects minus 2.

T(n,k) is also the dimension of the operad ComPreLie in arity n with k commutative products.

			Triangle T(n,k) begins:
n\k   0     1    2    3    4 ...
1     1;
2     2,    1;
3     9,    9,   1;
4    64,   96,  28,   1;
5   625, 1250, 625,  75,   1;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Cf. A000169 (k=0), A081131 (k=1).
Row sums are A052888.
Series reversion as e.g.f of A111492 with an offset of 1.

T(n) = my(x='x+O('x^(n+1))); [Vecrev(p) | p<-Vec(serlaplace( serreverse(log(1+x*y)*exp(-x)/y )))]
{my(A=T(10)); for(n=1, #A, print(A[n]))} \\ Andrew Howroyd, Oct 20 2023

E.g.f: series reversion in t of (log(1+x*t)/x)*exp(-t).
T(n,0) = n^(n-1).
T(n,n-1) = 1.

a(23) corrected by Andrew Howroyd, Jan 01 2024

A364708 Triangle of coefficient of the series reversion in t of the power series (exp(log(1+tx)/x)-1)exp(-t) as an e.g.f.

Original entry on oeis.org

1, 1, 1, 2, 6, 1, 6, 35, 22, 1, 24, 225, 310, 65, 1, 120, 1624, 3885, 1975, 171, 1, 720, 13132, 47929, 45080, 10367, 420, 1, 5040, 118124, 606060, 909489, 409416, 48034, 988, 1, 40320, 1172700, 7995455, 17445645, 13033398, 3152520, 204423, 2259, 1, 362880, 12753576, 110917400, 330281930, 369520305, 153751773, 21587950, 819120, 5065, 1

Offset: 1

Paul Laubie, Oct 20 2023

T(n,k) is also the dimension of the operad FMan in arity n with k commutative products.

The sum of each row is n^(n-1).

			Triangle T(n,k) begins:
n\k  0     1    2   3   4  ...
1    1;
2    1,    1;
3    2,    6,   1;
4    6,   35,  22,  1;
5   24,  225, 310, 65,  1;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Paul Laubie, Hypertrees and embedding of the FMan operad, arXiv:2401.17439 [math.QA], 2024.

Column k=0 is A000142(n-1).
Row sums are A000169.
Seems related to a signed version of A079510.

T(n) = my(x='x+O('x^(n+1))); [Vecrev(p) | p<-Vec(serlaplace( serreverse((exp(log(1+x*y)/y)-1)*exp(-x) )))]
{my(A=T(10)); for(n=1, #A, print(A[n]))} \\ Andrew Howroyd, Oct 20 2023

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

T(n,n-1) = 1.

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

Original entry on oeis.org

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

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User: Paul Laubie

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.

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A370360 Number of labeled semisimple rings with n elements.

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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.

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A370677 a(n) is the dimension of the multilinear part of the free flexible Lie-admissible algebra with n generators.

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A370161 a(n) is the dimension of the multilinear part of the free flexible algebra with n generators.

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A367753 Number of shapes of labeled forest of rooted hypertrees with n vertices.

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A367752 Number of shapes of labeled rooted hypertrees with n vertices.

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A364709 Triangle read by rows: T(n,k) is the number of forests of labeled rooted hypertrees with n vertices and weight k, 0 <= k < n.

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A364708 Triangle of coefficient of the series reversion in t of the power series (exp(log(1+t*x)/x)-1)*exp(-t) as an e.g.f.

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A364708 Triangle of coefficient of the series reversion in t of the power series (exp(log(1+tx)/x)-1)exp(-t) as an e.g.f.