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 A059615 #15 Jul 02 2025 16:02:00
%S A059615 0,1,15,240,6040,217365,10651011,681667840,55215038880,5521504648185,
%T A059615 668102052847735,96206695728917136,16258931576714668920,
%U A059615 3186750589054271109325,717018882536990087693835
%N A059615 a(n) is the number of non-parallel lines determined by a pair of vertices (extreme points) in the polytope of real n X n doubly stochastic matrices. The vertices are the n! permutation matrices.
%D A059615 M. Marcus, Hermitian Forms and Eigenvalues, in Survey of Numerical Analysis, J. Todd, ed. McGraw-Hill, New York, 1962.
%F A059615 a(n) = (1/2)*Sum_{k=0...n-2} binomial(n,k)^2 * (n-k)! * d(n-k) for n >= 2, where d(n) is the number of derangements of n elements: permutations of n elements with no fixed points - sequence A000166. Using the formula: d(n) = n!*Sum_{k=0..n} (-1)^k/k!, a(n) = (1/2)*Sum_{k=0..n-2} ((n!/k!)^2 * Sum_{m=0..n-k} (-1)^m/m!).
%e A059615 a(3) = 15 because there are 3! = 6 vertices and C(6,2) lines and in this case all are nonparallel so a(3) = C(6,2) = 15.
%p A059615 Digits := 200: with(combinat): d := n->n!*sum((-1)^j/j!,j=0..n): a059615 := n->1/2*sum( binomial(n,k)^2 * (n-k)!*d(n-k), k=0..n-2): for n from 1 to 30 do printf(`%d,`,round(evalf(a059615(n)))) od:
%o A059615 (PARI) a(n) = (1/2)*sum(k=0, n-2, ((n!/k!)^2 * sum(m=0, n-k, (-1)^m/m!))); \\ _Michel Marcus_, Mar 14 2018
%Y A059615 Cf. A000166, A059760.
%K A059615 nonn
%O A059615 1,3
%A A059615 Noam Katz (noamkj(AT)hotmail.com), Feb 18 2001
%E A059615 More terms from _James Sellers_, Feb 19 2001
%E A059615 Offset corrected by _Michel Marcus_, Mar 14 2018