A375853 Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.
2, 8, 8, 20, 56, 20, 40, 216, 216, 40, 70, 616, 1188, 616, 70, 112, 1456, 4576, 4576, 1456, 112, 168, 3024, 14040, 22880, 14040, 3024, 168, 240, 5712, 36720, 88400, 88400, 36720, 5712, 240, 330, 10032, 85272, 284240, 419900, 284240, 85272, 10032, 330
Offset: 2
Examples
Triangle begins: n\k 1 2 3 4 5 2: 2; 3: 8, 8; 4: 20, 56, 20; 5: 40, 216, 216, 40; 6: 70, 616, 1188, 616, 70; ...
Links
- Rebecca Bourn and Jeb F. Willenbring, Expected value of the one-dimensional earth mover's distance, Algebr. Stat. 11 (2020), no. 1, 53-78.
- Rebecca Bourn and William Q. Erickson, Proof of a conjecture of Bourn and Willenbring concerning a family of palindromic polynomials, arXiv:2307.02652 [math.CO], 2023.
- Colin Defant, Valentin Féray, Philippe Nadeau, and Nathan Williams, Wiener indices of minuscule lattices, Electron. J. Combin. 31 (2024), no.1, Paper No. 1.41, 23 pp.
- Ming-Jian Ding and Jiang Zeng, Some new results on minuscule polynomial of type A, arXiv:2308.16782 [math.CO], 2023.
- Eric Weisstein's World of Mathematics, Wiener Index.
Crossrefs
Programs
-
Maple
Trow := n -> seq(1/(4*n+2)*k*(n-k)*binomial(2*n+2, 2*k+1), k = 1..n-1): for n from 2 to 10 do Trow(n) od; # Alternatively, using the generating function of the row polynomials: rgf := (n, x) -> ((sqrt(x) - 1)^(2*n)*(2*n*sqrt(x) + x + 1) - (sqrt(x) + 1)^(2*n)*(-2*n*sqrt(x) + x + 1))/(16*sqrt(x)): T := (n, k) -> coeff(expand(rgf(n, x)), x, k): seq(print(seq(T(n, k), k = 1..n - 1)), n = 2..8): # Peter Luschny, Sep 22 2024
-
Mathematica
Flatten@Table[k*(n - k)*Binomial[2*n + 2, 2*k + 1]/(4*n + 2), {n, 2, 10}, {k, n - 1}] (* Zhining Yang, Sep 18 2024 *) -
PARI
T(n,k) = k*(n-k)*binomial(2*n+2,2*k+1)/(4*n+2) \\ Andrew Howroyd, Sep 01 2024
Formula
Sum_{k>=0} T(n, k) = A002699(n-1) (conjectured by Bourn and Erickson).
G.f.: T_n(x) = Sum_{k>=0} T(n, k)*x^k = (1 - x)^{2*n}*Sum_{k>=0}Sum_{alpha, beta} EMD_k(alpha, beta)*x^k, where EMD_k is the Earth Mover's Distance on (alpha, beta), and alpha, beta are the elements of composition of k into n parts.
T_n(x^2) = (n + 1)/8*((1 + x)^(2*n) + (1 - x)^(2*n)) - 1/(16*x)*((1 + x)^(2*n + 2) - (1 - x)^(2*n + 2)). (Proposition 3.1, arXiv:2308.16782)
Comments