A390893 - OEIS

%I #17 Nov 27 2025 22:03:04

%S 1,0,0,1,0,3,0,8,3,0,20,30,0,48,210,15,0,112,1260,315,0,256,6944,4200,

%T 105,0,576,36288,45360,3780,0,1280,182880,433440,81900,945,0,2816,

%U 897600,3825360,1386000,51975,0,6144,4316928,31944000,20207880,1663200,10395

%N Triangle read by rows: T(n,k) is the number of sets of noncrossing paths of size k that cover n nodes arranged in a circle with one node paths disallowed, 0 <= k <= floor(n/2).

%C Each path consists of straight line segments connecting two or more nodes on the circle. Each of the n nodes is used by exactly one path. Although each path is noncrossing, different paths are allowed to intersect. This differs from the noncrossing path sets studied in A390908.

%H Andrew Howroyd, <a href="/A390893/b390893.txt">Table of n, a(n) for n = 0..2600</a> (rows 0..100)

%F Column k is the inverse binomial transform of column k of A390896.

%F E.g.f.: exp(y*x*(exp(2*x) - 1)/4).

%F E.g.f. of column k: x^k*(exp(2*x) - 1)^k/(4^k*k!).

%F T(2*n,n) = A001147(n).

%e Triangle begins:

%e   1;

%e   0;

%e   0,    1;

%e   0,    3;

%e   0,    8,      3;

%e   0,   20,     30;

%e   0,   48,    210,      15;

%e   0,  112,   1260,     315;

%e   0,  256,   6944,    4200,     105;

%e   0,  576,  36288,   45360,    3780;

%e   0, 1280, 182880,  433440,   81900,   945;

%e   0, 2816, 897600, 3825360, 1386000, 51975;

%e   ...

%o (PARI) T(n) = { my(v=Vec(serlaplace( exp(y*x*(exp(2*x + O(x^n)) - 1)/4) ))); vector(#v, i, Vecrev(v[i], (i+1)\2)) }

%o { my(A=T(10)); for(i=1, #A, print(A[i])) }

%Y Row sums are A354323.

%Y Columns 0..4 are A000007, A001792(n-2), A332426, A359404, A360275.

%Y Cf. A001147, A390894 (with singleton paths), A390896 (not necessarily covering), A390897, A390908 (paths may not cross).

%K nonn,tabf

%O 0,6

%A _Andrew Howroyd_, Nov 23 2025