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A336245 Triangle read by rows: row n gives coefficients T(n,k), in descending powers of m, of a polynomial P_n(m) (of degree n - 1) in an expression for the number of bimonotone subdivisions B(m,n) of a grid with two rows.

%I A336245 #36 Jul 16 2020 02:28:35
%S A336245 1,1,0,1,3,-6,1,9,-4,-60,1,18,47,-258,-600,1,30,215,-270,-4896,-6720,
%T A336245 1,45,595,1455,-16796,-84660,-85680,1,63,1309,8925,-22526,-470148,
%U A336245 -1508424,-1239840,1,84,2506,30240,66409,-1500324,-11721396,-28649040,-20200320
%N A336245 Triangle read by rows: row n gives coefficients T(n,k), in descending powers of m, of a polynomial P_n(m) (of degree n - 1) in an expression for the number of bimonotone subdivisions B(m,n) of a grid with two rows.
%C A336245 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.
%C A336245 For m > n, the number of bimonotone subdivisions of P_(m,n) is given by B(m,n) = 2^(m-2)/(n-1)!*P_n(m), where P_n(m) is some monic polynomial with degree n - 1. See Theorem 1, p. 5, in Robeva and Sun (2020). (The authors' notation P_(m,n) for the grid should not be confused with their notation P_n(m) for the monic polynomial of degree n - 1 whose coefficients we tabulate here.)
%C A336245 We are not concerned here with the case m <= n. For more details, see A192933.
%H A336245 Elina Robeva and Melinda Sun, <a href="https://arxiv.org/abs/2007.00877">Bimonotone Subdivisions of Point Configurations in the Plane</a>, arXiv:2007.00877 [math.CO], 2020. See Table 1, p. 5.
%H A336245 Wikipedia, <a href="https://en.wikipedia.org/wiki/Faulhaber%27s_formula">Faulhaber's formula</a>.
%F A336245 B(m,n) = (2^(m-2)/(n-1)!) * Sum_{k=1..n} T(n,k)*m^(n-k).
%F A336245 B(m,n) = (2^(m-2)/(n-1)!) * P_n(m) = A192933(m,n).
%F A336245 B(m,n) = (2^(m-2)/(n-2)!) * (P_(n-1)(m) + S(m, n-2) - S(n-1, n-2) + Sum_{i=0..n-3} a_{i,n}*(S(m,i) - S(n-1,i))), where P_{n-1}(m) = m^(n-2) + Sum_{i=0..n-3} a_{i,n}*m^i and S(m,k) = Sum_{s=1..m} s^k. (Thus, a_{i,n} = T(n-1, n-1-i), and this formula is used in the PARI program below.)
%F A336245 B(m,n) = 2*(B(m,n-1) + B(m-1,n) - B(m-1,n-1)) for m > n.
%e A336245 Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
%e A336245   1;
%e A336245   1,  0;
%e A336245   1,  3,  -6;
%e A336245   1,  9,  -4,  -60;
%e A336245   1, 18,  47, -258,  -600;
%e A336245   1, 30, 215, -270, -4896, -6720;
%e A336245   ...
%e A336245 P_3(m) = m^2 + 3*m - 6.
%o A336245 (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
%o A336245 tabl(nn) = {my(p = 1, q); for (n=1, nn, if (n==1, q = p, q = (n-1)*(p + polf(n-2) - subst(polf(n-2), m, n-1) + sum(i=0, n-3, polcoef(p, i, m)*(polf(i)-subst(polf(i), m, n-1))))); print(Vec(q)); p = q;);}
%Y A336245 Cf. A192933, A336244.
%K A336245 sign,tabl
%O A336245 1,5
%A A336245 _Michel Marcus_ and _Petros Hadjicostas_, Jul 14 2020