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 A344216 #35 Jan 17 2024 10:34:42
%S A344216 1,3,16,104,768,6336,57888,581472,6379200,75977280,977045760,
%T A344216 13499930880,199537067520,3142504512000,52546707763200,
%U A344216 929908914278400,17366044153651200,341336836618444800,7044417438363648000
%N A344216 a(n) = n!*((n+1)/2 + 2*Sum_{k=2..n-1}(n-k)/(k+1)).
%C A344216 Conjecture: a(n) is the number of linear intervals in the weak order on the symmetric group S_n. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to a(7) = 57888.
%H A344216 Clément Chenevière, <a href="https://theses.hal.science/tel-04255439">Enumerative study of intervals in lattices of Tamari type</a>, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See p. 145.
%F A344216 From _Peter Luschny_, May 13 2021: (Start)
%F A344216 a(n) = (1/2) * n! * (4 * (n + 1) * H(n) - 9*n + 3), where H(n) are the harmonic numbers H(n) = A001008(n)/A002805(n).
%F A344216 a(n) = n! * [x^n] (3 - 8*x - 4*log(1 - x))/(2*(x - 1)^2).
%F A344216 a(n) = ((2*n^2 - 5*n - 1)*a(n-1) - (n^3 - 3*n^2 + 2*n)*a(n-2))/(n - 3) for n >= 4. (End)
%e A344216 For S_3, among the 17 intervals in the hexagon-shaped lattice, only the full lattice is not linear.
%p A344216 a := n -> (1/2)*n!*(4*(n + 1)*harmonic(n) - 9*n + 3):
%p A344216 # Or:
%p A344216 egf := (3 - 8*x - 4*ln(1 - x))/(2*(x - 1)^2):
%p A344216 ser := series(egf, x, 24): a := n -> n!*coeff(ser, x, n):
%p A344216 seq(a(n), n=1..19); # _Peter Luschny_, May 13 2021
%t A344216 Join[{1}, RecurrenceTable[{(n - 3) a[n] == (2 n^2 - 5 n - 1) a[n - 1] - (n^3 - 3 n^2 + 2 n) a[n - 2], a[2] == 3, a[3] == 16}, a, {n, 2, 19}]] (* _Peter Luschny_, May 13 2021 *)
%o A344216 (Sage)
%o A344216 def a(n):
%o A344216 return factorial(n)*((n+1)/2+2*sum((n-k)/(k+1) for k in range(2, n)))
%o A344216 (PARI) a(n) = n!*((n+1)/2+2*sum(k=2, n-1, (n-k)/(k+1))); \\ _Michel Marcus_, May 13 2021
%Y A344216 Cf. A344136, A344191, A344228 for similar sequences.
%Y A344216 Cf. A007767 for all intervals in the weak order on S_n.
%Y A344216 Cf. A001008, A002805.
%K A344216 nonn
%O A344216 1,2
%A A344216 _F. Chapoton_, May 13 2021