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

2,1

Comments

The T(n, k) are the coefficients of the minuscule polynomials of type A. They are the Wiener index of a minuscule lattice of type A, i.e., the Hasse diagram of the poset of order ideals in a k X (n - k) rectangle.

Links

Table of n, a(n) for n=2..46.

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.

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)

Example

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;
 ...

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 *)

Prog

(PARI) T(n, k) = k*(n-k)*binomial(2*n+2, 2*k+1)/(4*n+2) \\ Andrew Howroyd, Sep 01 2024

Crossrefs

Column 1 and main diagonal are A007290(n+1).

Row sums are A002699(n-1).

Half the sums of the gamma coefficients are A376072(n).

Sequence in context: A093907 A116471 A364294 * A146749 A250313 A180825

Adjacent sequences: A375850 A375851 A375852 * A375854 A375855 A375856

Keyword

nonn,easy,tabl

Author

Mingjian Ding, Aug 31 2024

Status

approved