A336245 -id:A336245 - cp's OEIS frontend

A192933 Triangle read by rows: T(n,k) = Sum_{i <= n, j <= k, (i,j) <> (n,k)} T(i,j), starting with T(1,1) = 1, for n >= 1 and 1 <= k <= n.

1, 1, 2, 2, 6, 12, 4, 16, 44, 88, 8, 40, 136, 360, 720, 16, 96, 384, 1216, 3152, 6304, 32, 224, 1024, 3712, 11296, 28896, 57792, 64, 512, 2624, 10624, 36416, 108032, 273856, 547712, 128, 1152, 6528, 29056, 109696, 362624, 1056896, 2661504, 5323008, 256, 2560, 15872, 76800, 314880, 1135616, 3659776, 10528768, 26380544, 52761088

Offset: 1

Andrea Raffetti, Jul 13 2011

The outer diagonal is A059435.
The second outer diagonal is A090442.
The third outer diagonal is essentially 2*A068766.
The first column is A011782.
The second column is essentially A057711 (not considering its first two terms).
The second column is essentially A129952 (not considering its first two terms).
The second column is essentially 2*A001792.
The differences between the terms of the second column is essentially 2*A045623.
The third column is essentially 4*A084266.
The cumulative sums of the third column are essentially 4*A176027.
T(n,k) = 0 for n < k. If this overriding constraint is not applied, you get A059576. - Franklin T. Adams-Watters, Jul 24 2011
For n >= 2 and 1 <= k <= n, T(n,k) is the number of bimonotone subdivisions of a 2-row grid with n points on the first row and k points on the second row (with the lower left point of the grid being the origin). A bimonotone subdivision of a convex polygon (the convex hull of the grid) is one where the internal dividing lines have nonnegative (including infinite) slopes. See Robeva and Sun (2020). - Petros Hadjicostas and Michel Marcus, Jul 15 2020

			Triangle (with rows n >= 1 and columns k = 1..n) begins:
   1;
   1,   2;
   2,   6,   12;
   4,  16,   44,    88;
   8,  40,  136,   360,   720;
  16,  96,  384,  1216,  3152,   6304;
  32, 224, 1024,  3712, 11296,  28896,  57792;
  64, 512, 2624, 10624, 36416, 108032, 273856, 547712;
  ...
Example: T(4,3) = 44 = 1 + 1 + 2 + 2 + 6 + 12 + 4 + 16.
From _Petros Hadjicostas_, Jul 15 2020: (Start)
Consider the following 2-row grid with n = 3 points at the top and k = 2 points at the bottom:
   A  B  C
   *--*--*
   |    /
   |   /
   *--*
   D  E
The sets of the dividing internal lines of the T(3,2) = 6 bimonotone subdivisions of the above 2-row grid are as follows: { }, {DC}, {DB}, {EB}, {DB, DC}, and {DB, EB}. We exclude subdivisions {EA} and {EA, EB} because they have at least one dividing line with a negative slope. (End)

Andrea Raffetti, Rows n = 1..13 of triangle
Elina Robeva and Melinda Sun, Bimonotone Subdivisions of Point Configurations in the Plane, arXiv:2007.00877 [math.CO], 2020.

Cf. A001003, A001792, A006318, A011782, A045623, A057711, A059435, A059576, A068766, A084266, A090442, A129952, A166444, A176027, A336245.

lista(nn) = {my(T=matrix(nn, nn)); T[1,1] = 1; for (n=2, nn, for (k=1, n, T[n,k] = sum(i=1, n, sum(j=1, k, if ((i!=n) || (j!=k), T[i,j]))););); vector(nn, k, vector(k, i, T[k, i]));} \\ Michel Marcus, Mar 18 2020

T(n,1) = 2^(n-2) for n >= 2.
T(n,2) = n*2^(n-2) for n >= 2.
T(n,3) = 2^(n-2)*((n-k+1)^2 + 7*(n-k+1) + 4)/2 = 2^(n-3)*(n^2 + 3*n - 6) for k = 3 and n >= 3.
In general: For 1 <= k <= n with (n,k) <> 1,
T(n,k) = 2^(n-2)*Sum_{i=0..k-1} c(k,i)*(n-k+1)^(k-1-i)/(k-1)! and
T(n,k) = 2^(n-2)*Sum_{j=0..k-1} c(k,k-1-j)*(n-k+1)^j/(k-1)!
with c(k,i) being specific coefficients. Below are the first values for c(k,i):
  1;
  1,  1;
  1,  7,    4;
  1, 18,   77,   36;
  1, 34,  359, 1238,   528,
  1, 55, 1065, 8705, 26654, 10800;
... [Formula corrected by Petros Hadjicostas, Jul 15 2020]
The diagonal of this triangle for c(k,i) divided by (k-1)! (except for the first term) is equal to the Shroeder number sequence A006318(k-1).
From Petros Hadjicostas and Michel Marcus, Jul 15 2020: (Start)
T(n,1) = 2^(n-2) for n >= 2; T(n,k) = 2*(T(n,k-1) + T(n-1,k) - T(n-1,k-1)) for n > k >= 2; T(n,n) = 2*T(n,n-1) for n = k >= 2; and T(n,k) = 0 for 1 <= n < k. [Robeva and Sun (2020)] (They do not specify T(1,1) explicitly since they do not care about subdivisions of a degenerate polygon with only one side.)
T(n,k) = (2^(n-2)/(k-1)!) * P_k(n) = (2^(n-2)/(k-1)!) * Sum_{j=1..k} A336245(k,j)*n^(k-j) for n >= k >= 1 with (n,k) <> (1,1), where P_k(n) is some polynomial with integer coefficients of degree k-1. [Robeva and Sun (2020)]
A336245(k,j) = Sum_{s=0..j-1} c(k,s) * binomial(k-1-s, k-j) * (1-k)^(j-1-s) for 1 <= j <= k, in terms of the above coefficients c(k,i).
So c(k,s) = Sum_{j=1..s+1} A336245(k,j) * binomial(k-j, k-s-1) * (k-1)^(s+1-j) for k >= 1 and 0 <= s <= k-1, obtained by inverting the binomial transform.
Bivariate o.g.f.: x*y*(1 - x)*(1 - 2*y*g(2*x*y))/(1 - 2*x - 2*y + 2*x*y), where g(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = g.f. of A001003.
Letting y = 1 in the above joint o.g.f., we get the o.g.f. of the row sums: x*(1-x)*(2*g(2*x) - 1). It can then be easily proved that
Sum_{k=1..n} T(n,k) = 2^n*A001003(n-1) - 2^(n-1)*A001003(n-2) for n >= 3. (End)

Offset changed by Andrew Howroyd, Dec 31 2017

Name edited by Petros Hadjicostas, Jul 15 2020

A336244 Triangle read by rows: row n gives coefficients T(n,k), in descending powers of m, of a polynomial Q_n(m) (of degree n - 1) in an expression for the number of subdivisions A(m,n) of a grid with two rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 2, 1, 12, 29, 6, 1, 22, 131, 206, 24, 1, 35, 385, 1525, 1774, 120, 1, 51, 895, 6585, 19624, 18204, 720, 1, 70, 1792, 21070, 117019, 281260, 218868, 5040, 1, 92, 3234, 55496, 492849, 2210348, 4483436, 3036144, 40320

Offset: 1

Michel Marcus and Petros Hadjicostas, Jul 14 2020

Let P_(m,n) denote a grid with 2 rows that has m points in the top row and n points in the bottom, aligned at the left, and let the bottom left point be at the origin.
For m > n, the number of subdivisions of P_(m,n) is given by A(m,n) = 2^(m-2)/(n-1)!*Q_n(m), where Q_n(m) is some monic polynomial of degree n-1. See Theorem 2, p. 6, in Robeva and Sun (2020).
By symmetry, A(m,n) = A(n,m). For more information and formulas, see A059576.

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins
  1;
  1,  1;
  1,  5,   2;
  1, 12,  29,   6;
  1, 22, 131, 206, 24;
  ...
Q_3(m) = m^2 + 5*m + 2.

Elina Robeva and Melinda Sun, Bimonotone Subdivisions of Point Configurations in the Plane, arXiv:2007.00877 [math.CO], 2020. See Table 2, p. 5.
Wikipedia, Faulhaber's formula.

Cf. A059576, A103213, A336245.

# We assume the rows indexed by the degree of the polynomials, n = 0,1,2,...
A336244row := proc(n) local p, k, s, b; p := 1;
b := n -> bernoulli(n, x+1) - bernoulli(n, 1);
for k from 1 to n-1 do
  s := p + add(coeff(p, x, i-1)*b(i)/i, i=1..k-1);
  p := b(k) + k*s od;
seq(coeff(p, x, n-i), i=1..n) end:
seq(A336244row(n), n=0..9); # Peter Luschny, Jul 15 2020

b[n_] := BernoulliB[n, x + 1] - BernoulliB[n, 1]; b[1] = x;
row[n_] := Module[{p = 1, s}, Do[s = p + Sum[Coefficient[p, x, i-1] b[i]/i, {i, 1, k-1}]; p = b[k] + k s, {k, 1, n-1}]; CoefficientList[p, x] // Reverse];
row /@ Range[9] // Flatten (* Jean-François Alcover, Aug 21 2020, after Peter Luschny *)

polf(n) = if (n==0, return(m)); my(p=bernpol(n+1,m)); (subst(p, m, m+1) - subst(p, m, 0))/(n+1);  \\ Faulhaber
tabl(nn) = {my(p = 1, q); for (n=1, nn, if (n==1, q = p, q = (n-1)*(p + polf(n-2) + sum(i=0, n-3, polcoef(p, i, m)*polf(i)))); print(Vec(q)); p = q;);}

A(m,n) = (2^(m-2)/(n-1)!) * Sum_{k=1..n} T(n,k)*m^(n-k).
A(m,n) = (2^(m-2)/(n-1)!) * Q_n(m) = A059576(m-1, n-1) (provided the latter is viewed as a square array rather than a triangle).
A(m,n) = (2^(m-2)/(n-2)!) * (Q_(n-1)(m) + Sum_{i=1..m} Q_(n-1)(i)).
A(m,n) = 2*(A(m,n-1) + A(m-1,n) - A(m-1,n-1)) for m > n.
T(n, 1) = 1 and T(n, n) = (n - 1)!.
Conjectures:
(a) T(n,2) = (n - 1)*(3*n - 4)/2.
(b) T(n,3) = (n - 2)*(n - 1)*(27*n^2 - 97*n + 72)/24.
(c) T(n,4) = (n - 3)*(n - 2)*(n - 1)^2*(27*n^2 - 156*n + 208)/48.
(d) T(n, n - 1) = (n - 1)!*Sum_{k=1..n-1} binomial(n-1, k)/k = A103213(n-1).

cp's OEIS Frontend

A192933 Triangle read by rows: T(n,k) = Sum_{i <= n, j <= k, (i,j) <> (n,k)} T(i,j), starting with T(1,1) = 1, for n >= 1 and 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions