A103505 -id:A103505 - cp's OEIS frontend

A139331 Triangle read by rows: ConvOffsStoT transform of A103505.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 12, 6, 1, 1, 12, 72, 72, 12, 1, 1, 20, 240, 720, 240, 20, 1, 1, 30, 600, 3600, 3600, 600, 30, 1, 1, 56, 1680, 16800, 33600, 16800, 1680, 56, 1, 1, 72, 4032, 60480, 201600, 201600, 60480, 4032, 72, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Apr 13 2008

			First few rows of the triangle are:
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 6, 12, 6, 1;
1, 12, 72, 72, 12, 1;
1, 20, 240, 720, 240, 20, 1;
1, 30, 600, 3600, 3600, 600, 30, 1;
...
Row 4 = (1, 6, 12, 6, 1) since ConvOffs transform of (1, 1, 2, 6) = (1, 6, 12, 6, 1); where A103505 = (1, 1, 2, 6, 12, 20, 30, 56, 72,...).

Cf. A103505.

A279019 Least possible number of diagonals of simple convex polyhedron with n faces.

Original entry on oeis.org

0, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450

Offset: 4

Vladimir Letsko, Dec 03 2016

Obviously, a pyramid has no diagonals. Hence minimum of diagonals of an arbitrary convex polyhedron having n faces is equal to 0.
Minimum number of diagonals among simple convex polyhedra having n faces is obtained from a polyhedron with two triangular faces, n-4 quadrangular faces and two (n-1)-sided faces. A polyhedron having 3 triangular faces, 3 pentagonal faces and 1 hexagonal face gives another example of a simple convex polyhedron with the least possible number of diagonals for n = 7. A polyhedron having 4 triangular faces and 4 hexagonal faces gives a similar example for n = 8.
Essentially the same as A103505 and A002378. - R. J. Mathar, Dec 05 2016

Colin Barker, Table of n, a(n) for n = 4..1000
David Bremner and Victor Klee, Inner Diagonals of Convex Polytopes, Journal of Combinatorial Theory, Series A, Volume 87, Issue 1, July 1999, Pages 175-197.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000944, A002378, A279015, A279022, A007531.

Table[(n-4)(n-5),{n,4,60}] (* or *) LinearRecurrence[{3,-3,1},{0,0,2},60] (* Harvey P. Dale, Sep 23 2019 *)

concat(vector(2), Vec(2*x^6 / (1 - x)^3 + O(x^60))) \\ Colin Barker, Dec 05 2016

a(n) = n^2 - 9*n + 20 = (n-4)*(n-5).
G.f.: -2*x^6/(x-1)^3. - R. J. Mathar, Dec 05 2016
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6. - Colin Barker, Dec 05 2016
E.g.f.: exp(x)*(20 - 8*x + x^2) - x^3/3 - 3*x^2 - 12*x - 20. - Stefano Spezia, Nov 24 2019
From Amiram Eldar, Jul 09 2023: (Start)
Sum_{n>=6} 1/a(n) = 1.
Sum_{n>=6} (-1)^n/a(n) = 2*log(2) - 1. (End)

A236532 Triangle T(n,k) read by rows: T(n,k) = floor(n*k/(n+k)), with 1 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 2, 0, 1, 2, 2, 2, 3, 0, 1, 2, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 3, 4, 0, 1, 2, 2, 3, 3, 3, 4, 4, 0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 0, 1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 0, 1, 2, 3, 3, 4, 4, 4, 5, 5

Offset: 1

Alex Ratushnyak, Jan 27 2014

It appears that the least m such that T(m, n) = n-1 is given by A103505(n) for n>= 1. - Michel Marcus, Feb 25 2020

			Triangle begins:
  0
  0 1
  0 1 1
  0 1 1 2
  0 1 1 2 2
  0 1 2 2 2 3
  0 1 2 2 2 3 3
  0 1 2 2 3 3 3 4
  0 1 2 2 3 3 3 4 4
  0 1 2 2 3 3 4 4 4 5
  0 1 2 2 3 3 4 4 4 5 5
  0 1 2 3 3 4 4 4 5 5 5 6
  0 1 2 3 3 4 4 4 5 5 5 6 6
  0 1 2 3 3 4 4 5 5 5 6 6 6 7
  0 1 2 3 3 4 4 5 5 6 6 6 6 7 7
  0 1 2 3 3 4 4 5 5 6 6 6 7 7 7 8
  0 1 2 3 3 4 4 5 5 6 6 7 7 7 7 8 8
  0 1 2 3 3 4 5 5 6 6 6 7 7 7 8 8 8 9

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

Cf. A004526, A103505.

T(n,k)={n*k\(n+k)} \\ Andrew Howroyd, Feb 24 2020

for n in range(1, 21):
  for k in range(1, n+1):
    print(n*k // (n+k), end=', ')
  #print() # Uncomment to display as a triangle

T(n, n) = floor(n/2). See A004526. - Michel Marcus, Feb 25 2020

A359365 a(n) = lcm([ n!*binomial(n-1, m-1) / m! for m = 1..n ]) with a(0) = 1.

Original entry on oeis.org

1, 1, 2, 6, 72, 240, 3600, 75600, 1411200, 10160640, 457228800, 4191264000, 184415616000, 2054916864000, 12466495641600, 1308982042368000, 314155690168320000, 14241724620963840000, 2178983867007467520000, 37260624125827694592000, 337119932567012474880000

Offset: 0

Peter Luschny, Dec 30 2022

nonn

The lcm of the rows of the unsigned Lah triangle (for k >= 1).

Michael De Vlieger, Table of n, a(n) for n = 0..390

Cf. A271703 (unsigned Lah numbers), A103505 (gcd counterpart).

# Maple has the convention integer lcm() = 1, which covers the case n = 0.
a := n -> ilcm(seq(n!*binomial(n-1, m-1) / m!, m = 1..n)):
seq(a(n), n = 0..20);

{1}~Join~Table[LCM @@ Array[n!*Binomial[n - 1, # - 1]/#! &, n], {n, 20}] (* Michael De Vlieger, Dec 30 2022 *)

a(n) = lcm(vector(n, m, n!*binomial(n-1, m-1) / m!)); \\ Michel Marcus, Dec 30 2022

from functools import cache
from sympy import lcm
def A359365 (n: int) -> int:
    @cache
    def l(n: int) -> list[int]:
        if n == 0: return [1]
        row: list[int] = l(n - 1) + [1]
        row[0] = 0
        for k in range(n - 1, 0, -1):
            row[k] = row[k] * (n + k - 1) + row[k - 1]
        return row
    return lcm(l(n)[1:])
print([A359365(n) for n in range(21)])

A358185 Coefficients of x^n/n! in the expansion of (1 - x)*log(1 - x).

Original entry on oeis.org

0, -1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000, 25852016738884976640000

Offset: 0

Wolfdieter Lang, Nov 14 2022

The negated sequence gives the compositional inverse of 1 - exp(W(-x)) with the principal branch of Lambert's W function.

See A000312 for the two formulas of Vladimir Kruchinin, Sep 15 2010, and Peter Bala, Dec 09 2011. Here stated the other way around.

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], (2018) eq (60).

Cf. A000142, A103505, A000312, A159333, A104150.

With[{m = 25}, Range[0, m]! * CoefficientList[Series[(1-x) * Log[1-x], {x, 0, m}], x]] (* Amiram Eldar, Nov 14 2022 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)*log(1-x)), -N) \\ Michel Marcus, Sep 16 2023

E.g.f.: (1 - x) * log(1 - x).

a(0) = 0, a(1) = -1, a(n) = (n-2)! = A000142(n-2), for n >= 2.

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A139331 Triangle read by rows: ConvOffsStoT transform of A103505.

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A279019 Least possible number of diagonals of simple convex polyhedron with n faces.

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A236532 Triangle T(n,k) read by rows: T(n,k) = floor(n*k/(n+k)), with 1 <= k <= n.

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A359365 a(n) = lcm([ n!*binomial(n-1, m-1) / m! for m = 1..n ]) with a(0) = 1.

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Maple

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A358185 Coefficients of x^n/n! in the expansion of (1 - x)*log(1 - x).

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