OFFSET
1,5
COMMENTS
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.
LINKS
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.
FORMULA
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).
EXAMPLE
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.
MAPLE
# 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
MATHEMATICA
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 *)
PROG
(PARI) 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; ); }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Michel Marcus and Petros Hadjicostas, Jul 14 2020
STATUS
approved