This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A375853 #44 Mar 30 2025 06:32:20
%S A375853 2,8,8,20,56,20,40,216,216,40,70,616,1188,616,70,112,1456,4576,4576,
%T A375853 1456,112,168,3024,14040,22880,14040,3024,168,240,5712,36720,88400,
%U A375853 88400,36720,5712,240,330,10032,85272,284240,419900,284240,85272,10032,330
%N 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.
%C A375853 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.
%H A375853 Rebecca Bourn and Jeb F. Willenbring, <a href="https://doi.org/10.2140/astat.2020.11.53">Expected value of the one-dimensional earth mover's distance</a>, Algebr. Stat. 11 (2020), no. 1, 53-78.
%H A375853 Rebecca Bourn and William Q. Erickson, <a href="https://arxiv.org/abs/2307.02652">Proof of a conjecture of Bourn and Willenbring concerning a family of palindromic polynomials</a>, arXiv:2307.02652 [math.CO], 2023.
%H A375853 Colin Defant, Valentin Féray, Philippe Nadeau, and Nathan Williams, <a href="https://doi.org/10.37236/12002">Wiener indices of minuscule lattices</a>, Electron. J. Combin. 31 (2024), no.1, Paper No. 1.41, 23 pp.
%H A375853 Ming-Jian Ding and Jiang Zeng, <a href="https://arxiv.org/abs/2308.16782">Some new results on minuscule polynomial of type A</a>, arXiv:2308.16782 [math.CO], 2023.
%H A375853 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%F A375853 Sum_{k>=0} T(n, k) = A002699(n-1) (conjectured by Bourn and Erickson).
%F A375853 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.
%F A375853 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)
%e A375853 Triangle begins:
%e A375853 n\k 1 2 3 4 5
%e A375853 2: 2;
%e A375853 3: 8, 8;
%e A375853 4: 20, 56, 20;
%e A375853 5: 40, 216, 216, 40;
%e A375853 6: 70, 616, 1188, 616, 70;
%e A375853 ...
%p A375853 Trow := n -> seq(1/(4*n+2)*k*(n-k)*binomial(2*n+2, 2*k+1), k = 1..n-1):
%p A375853 for n from 2 to 10 do Trow(n) od;
%p A375853 # Alternatively, using the generating function of the row polynomials:
%p A375853 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)):
%p A375853 T := (n, k) -> coeff(expand(rgf(n, x)), x, k):
%p A375853 seq(print(seq(T(n, k), k = 1..n - 1)), n = 2..8): # _Peter Luschny_, Sep 22 2024
%t A375853 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 *)
%o A375853 (PARI) T(n,k) = k*(n-k)*binomial(2*n+2,2*k+1)/(4*n+2) \\ _Andrew Howroyd_, Sep 01 2024
%Y A375853 Column 1 and main diagonal are A007290(n+1).
%Y A375853 Row sums are A002699(n-1).
%Y A375853 Half the sums of the gamma coefficients are A376072(n).
%K A375853 nonn,easy,tabl
%O A375853 2,1
%A A375853 _Mingjian Ding_, Aug 31 2024