Harry Richman - cp's OEIS frontend

A366040 Irregular triangle read by rows: T(n,k) = number of cells of dimension k in the tropical Schottky locus of genus n.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 2, 2, 1, 1, 1, 2, 3, 5, 9, 12, 15, 17, 15, 7, 4

Offset: 2

Harry Richman, Oct 23 2023

Row n has 3n + 1 entries, for 0 <= k <= 3n.

			The irregular triangle T(n,k) begins:
n\k  0  1  2  3  4  5  6   7   8   9  10 11 12...
2:   1, 1, 1, 1;
3:   1, 1, 1, 2, 2, 1, 1;
4:   1, 1, 1, 2, 3, 4, 5,  4,  2,  2;
5:   1, 1, 1, 2, 3, 5, 9, 12, 15, 17, 15, 7, 4;
...

Melody Chan, Combinatorics of the tropical Torelli map, Algebra Number Theory, 6 (2012), 1133-1169.

Row sums are A220444.

A366039 Irregular triangle read by rows: T(n,k) = number of cells of dimension k in the moduli space of tropical curves of genus n.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 5, 9, 12, 8, 5, 1, 3, 7, 21, 43, 75, 89, 81, 42, 17, 1, 3, 11, 34, 100, 239, 492, 784, 1002, 926, 632, 260, 71

Offset: 2

Harry Richman, Oct 23 2023

Row n has 3n + 1 entries, for 0 <= k <= 3n.

			The irregular triangle T(n,k) begins:
n\k  0   1   2   3    4    5    6    7     8    9   10   11  12...
2:   1,  2,  2,  2;
3:   1,  2,  5,  9,  12,   8,   5;
4:   1,  3,  7, 21,  43,  75,  89,  81,   42,  17;
5:   1,  3, 11, 34, 100, 239, 492, 784, 1002, 926, 632, 260, 71;
...

Melody Chan, Combinatorics of the tropical Torelli map, Algebra Number Theory, 6 (2012), 1133-1169.

Last entry in each row is A005967.

Row sums are A174224.

A364441 Number of facets of the balanced minimum evolution polytope on n species.

Original entry on oeis.org

3, 52, 90262

Offset: 4

Harry Richman, Jul 24 2023

The balanced minimum evolution (BME) polytope of order n is a polytope in binomial(n, 2)-dimensional space whose vertices are indexed by bifurcating phylogenetic trees on n species. The coordinates of the ambient space are indexed by pairs (i, j) of distinct species, and at a vertex corresponding to tree T, the value at coordinate (i, j) is equal to 2^-k(i, j) where k(i, j) is the number of internal nodes on the path from i to j in T.

Stefan Forcey, Encyclopedia of Combinatorial Polytope Sequences: Balanced Minimum Evolution Polytope.
Stefan Forcey, Logan Keefe, and William Sands, Split-Facets for Balanced Minimum Evolution Polytopes and the Permutoassociahedron, Bull. Math. Biol., 79 (2017), 975-994.

Cf. A364505.

A364505 T(n, k) = number of k-dimensional faces in the BME polytope on n species, 0 <= k <= binomial(n, 2) - n.

Original entry on oeis.org

3, 3, 1, 15, 105, 250, 210, 52, 1, 105, 5460, 105945, 635265, 1715455, 2373345, 1742445, 640140, 90262, 1

Offset: 4

Harry Richman, Jul 26 2023

The balanced minimum evolution (BME) polytope of order n is the convex hull of the BME vectors of all phylogenetic trees on n species. The BME polytope of order n has dimension binomial(n, 2) - n.

			Table begins:
    3,    3,      1;
   15,  105,    250,    210,      52,       1;
  105, 5460, 105945, 635265, 1715455, 2373345, 1742445, 640140, 90262, 1;

Maria Angelica Cueto and Frederick A. Matsen, Polyhedral geometry of phylogenetic rogue taxa, Bull. Math. Biol., 73 (2011), 1202-1226.
K. Eickmeyer, P. Huggins, L. Pachter, and R. Yoshida, On the optimality of the neighbor-joining algorithm, Algorithms Mol Biol. 3 (2008), Article number 5.
Stefan Forcey, Balanced Minimum Evolution Polytope, Encyclopedia of Combinatorial Polytope Sequences (Hedra Zoo).

First column T(n, 0) is A001147.

Next-to-last entry T(n, binomial(n, 2) - n - 1) in each row is A364441.

A363273 Irregular triangle read by rows: T(n,k) = number of unlabeled binary rooted trees with n leaves, where both children have at least k leaves, 1 <= k <= n/2.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 6, 3, 1, 11, 5, 2, 23, 12, 6, 3, 46, 23, 12, 6, 98, 52, 29, 18, 6, 207, 109, 63, 40, 18, 451, 244, 146, 100, 54, 21, 983, 532, 325, 227, 135, 66, 2179, 1196, 745, 538, 342, 204, 66, 4850, 2671, 1688, 1237, 823, 529, 253, 10905, 6055, 3876, 2893, 1991, 1370, 782, 276

Offset: 2

Harry Richman, May 24 2023

			Table begins:
  1;
  1;
  2,   1;
  3,   1;
  6,   3,  1;
 11,   5,  2;
 23,  12,  6,  3;
 46,  23, 12,  6;
 98,  52, 29, 18,  6;
207, 109, 63, 40, 18;
...

Andrew Howroyd, Table of n, a(n) for n = 2..2501 (rows 1..100)

First column k = 1 is A001190.
Sums along upwards diagonals are A000671.
Cf. A363272.

T(n)={my(A=vector(n), R=vector(n)); A[1]=1; R[1]=[]; for(i=2, n, my(t=vector(i\2, j, if(2*jAndrew Howroyd, Jan 01 2024

T(n,k) = Sum_{j >= k} A363272(n,j).

Sum_{k >= 1} T(n-k, k) = A000671(n-2).

Terms a(27) and beyond from Andrew Howroyd, Jan 01 2024

A363272 Irregular triangle read by rows: T(n,k) = number of unlabeled binary rooted trees with n leaves, where some child tree has k leaves, 1 <= k <= n/2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 1, 6, 3, 2, 11, 6, 3, 3, 23, 11, 6, 6, 46, 23, 11, 12, 6, 98, 46, 23, 22, 18, 207, 98, 46, 46, 33, 21, 451, 207, 98, 92, 69, 66, 983, 451, 207, 196, 138, 138, 66, 2179, 983, 451, 414, 294, 276, 253, 4850, 2179, 983, 902, 621, 588, 506, 276

Offset: 2

Harry Richman, May 24 2023

			Table begins:
 1;
 1;
 1,  1;
 2,  1;
 3,  2,  1;
 6,  3,  2;
11,  6,  3,  3;
23, 11,  6,  6;
46, 23, 11, 12,  6;
98, 46, 23, 22, 18;
...

Andrew Howroyd, Table of n, a(n) for n = 2..2501 (rows 2..100)

Row sums are A001190.
First column k=1 is T(n,1) = A001190(n-1).
Cf. A000671, A363273.

T(n)={my(A=vector(n), R=vector(n)); A[1]=1; R[1]=[]; for(i=2, n, R[i] = vector(i\2, j, if(2*jAndrew Howroyd, Jan 01 2024

T(n,k) = A001190(k) * A001190(n-k) if k < n/2; otherwise
T(2k,k) = A001190(k) * (A001190(k) + 1) / 2 = A000217(A001190(n)).
Sum_{k >= 1} T(n,k) = A001190(n).
Sum_{i >= k} T(n,i) = A363273(n,k).
Sum_{i <= n-1, i+j >= n} T(i,j) = A000671(n-2).

Terms a(32) and beyond from Andrew Howroyd, Jan 01 2024

A363257 a(n) = floor( ((a(n-1) + 1) / 2)^2 ) + 1 for n >= 1, with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 31, 257, 16642, 69247363, 1198799355237125, 359279973529237254190922184970, 32270524844792355518177347536627638351478874995525184567711

Offset: 0

Harry Richman, May 23 2023

nonn

Iterated application of A033638, with a shift.

Andrew Howroyd, Table of n, a(n) for n = 0..16
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

Cf. A003095, A006894, A014980, A033638.

a(n) = if(n < 1, 0, floor( ((a(n-1) + 1) / 2)^2 ) + 1) \\ Andrew Howroyd, Jan 01 2024

a(n) = A033638(a(n-1)+1) for n > 0.

log a(n) ~ C * 2^n for some constant C.

A360079 Finite differences of Moebius function for the floor quotient poset.

Original entry on oeis.org

1, -2, 0, 1, 0, 1, 0, -1, 1, 0, 0, -1, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2

Offset: 1

Harry Richman, Jan 24 2023

sign

a(n) = mu(n) - mu(n-1), where mu(n) = A360078(n) is the Moebius function of the floor quotient poset.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
J.-P. Cardinal, Symmetric matrices related to the Mertens function, arXiv:0811.3701 [math.NT], 2008.
J. C. Lagarias and D. H. Richman, The floor quotient partial order, arXiv:2212.11689 [math.NT], 2022.

Cf. A002321, A008683, A360078.

isFQ d n = (n `div` (n `div` d)) == d
fqMobius 1 = 1
fqMobius n = - sum [fqMobius d | d <- [1..(n-1)], d `isFQ` n]
a360079 1 = 1
a360079 n = fqMobius n - fqMobius (n-1)
-- Harry Richman, Jun 13 2025

LinearSolve[Table[If[Floor[i/j] > Floor[i/(j + 1)], 1, 0], {i, n}, {j, n}] . Table[If[i >= j, 1, 0], {i, n}, {j, n}], UnitVector[n, 1]]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, my(S=Set(vector(n-1, k, n\(k+1)))); v[n]=-sum(i=1, #S, v[S[i]])); vector(#v, i, v[i]-if(i>1, v[i-1]))} \\ Andrew Howroyd, Jan 24 2023

A360078 Moebius function for the floor quotient poset.

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, -1, -1, -1, -1, -1, -1, 0, 0, 0, -1, -1, -1, -2, -2, -2, -2, -2, -2, -1, -1, -1, -1, 0, 0, -1, -1, -1, -2, -2, -2, -2, -3, -3, -3, -3, -3, -1, -1, -1, -1, -1, -1, 1, 1, 1, 0, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Harry Richman, Jan 24 2023

Say d is a "floor quotient" of n if d = [n/k] for some positive integer k. This defines a partial order relation on the positive integers. This sequence records the Moebius function values of this poset.

			For n = 9, the set of floor quotients of 9 are Q(9) = {1, 2, 3, 4, 9} with Moebius values a(1) = 1, a(2) = -1, a(3) = -1, and a(4) = 0. The Moebius recursion requires that the Moebius values summed over Q(9) must equal zero, so a(9) = 1.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
J.-P. Cardinal, Symmetric matrices related to the Mertens function, arXiv:0811.3701 [math.NT], 2008-2009.
J. C. Lagarias and D. H. Richman, The floor quotient partial order, Adv. Appl. Math., 153 (2024); arXiv:2212.11689 [math.NT], 2022-2023.

Cf. A002321, A008683, A360079.

isFQ d n = (n `div` (n `div` d)) == d
a360078 1 = 1
a360078 n = - sum [a360078 d | d <- [1..(n-1)], d `isFQ` n]
-- Harry Richman, Jun 13 2025

LinearSolve[Table[If[Floor[i/j] > Floor[i/(j + 1)], 1, 0], {i, n}, {j, n}], UnitVector[n, 1]]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, my(S=Set(vector(n-1, k, n\(k+1)))); v[n]=-sum(i=1, #S, v[S[i]])); v} \\ Andrew Howroyd, Jan 24 2023

A351167 Partial sums of A350682.

Original entry on oeis.org

1, 0, 0, -1, -1, -1, -2, -2, -2, -3, -3, -3, -4, -4, -3, -4, -4, -4, -4, -3, -3, -4, -4, -3, -4, -4, -4, -5, -5, -5, -6, -6, -6, -7, -7, -7, -8, -8, -7, -7, -7, -7, -8, -6, -6, -7, -7, -6, -7, -7, -7, -8, -8, -7, -5, -4, -4, -5, -4, -3, -4, -4, -3, -3, -2, -2, -3, -3, -3, -4, -4, -4, -5, -5, -4, -5, -4, -4, -4, -4, -4, -5, -5, -4, -5, -5, -5, -6, -6, -5, -5, -5, -5, -6, -6, -6, -7, -7, -7, -7

Offset: 1

Rohan Pandey, Harry Richman, Feb 03 2022

sign

Partial sums of Möbius values of triangular numbers under divisibility relation.

Rohan Pandey and Harry Richman, The Möbius function of the poset of triangular numbers under divisibility, arXiv:2402.07934 [math.NT], 2024. See pp. 2, 8.

Cf. A000217, A002321, A350682.

Accumulate@ With[{m = 100}, LinearSolve[Table[If[Mod[i (i + 1), j (j + 1)] == 0, 1, 0], {i, m}, {j, m}], UnitVector[m, 1]]] (* Michael De Vlieger, Feb 04 2022, after Harry Richman at A350682 *)

lista(nn) = {my(v=vector(nn, k, k*(k+1)/2)); my(m=matrix(nn, nn, n, k, ! (v[n] % v[k]))); m = 1/m; my(w = vector(nn, k, m[k, 1])); vector(nn-1, k, sum(i=1, k, w[i]));} \\ Michel Marcus, Feb 16 2022

from sympy import *
triangular_numbers = ([(x * (x + 1) // 2) for x in range(1, 101)])
def Mobius_Matrix(lst):
    zeta_array = [[0 if n % m != 0 else 1 for n in lst] for m in lst]
    return Matrix(zeta_array) ** -1
M = Mobius_Matrix(triangular_numbers)
N = M[0, :].tolist()
def sum_function(lst):
    sum_list = [sum(lst[:i+1]) for i in range(len(lst))]
    return sum_list
S = sum_function(N[0])
print(S)

cp's OEIS Frontend

User: Harry Richman

A366040 Irregular triangle read by rows: T(n,k) = number of cells of dimension k in the tropical Schottky locus of genus n.

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A366039 Irregular triangle read by rows: T(n,k) = number of cells of dimension k in the moduli space of tropical curves of genus n.

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A364441 Number of facets of the balanced minimum evolution polytope on n species.

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A364505 T(n, k) = number of k-dimensional faces in the BME polytope on n species, 0 <= k <= binomial(n, 2) - n.

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A363273 Irregular triangle read by rows: T(n,k) = number of unlabeled binary rooted trees with n leaves, where both children have at least k leaves, 1 <= k <= n/2.

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A363272 Irregular triangle read by rows: T(n,k) = number of unlabeled binary rooted trees with n leaves, where some child tree has k leaves, 1 <= k <= n/2.

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A363257 a(n) = floor( ((a(n-1) + 1) / 2)^2 ) + 1 for n >= 1, with a(0) = 0.

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A360079 Finite differences of Moebius function for the floor quotient poset.

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A360078 Moebius function for the floor quotient poset.

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