cp's OEIS Frontend

A244312 Triangle read by rows: T(n,k) is the number of single loop solutions formed by n proper arches (connecting odd vertices to even vertices in the range 1 to 2n) above the x axis, k of which connect an odd vertex to a higher even vertex, with a rainbow of n arches below the x axis.

%I A244312 #33 May 07 2025 11:31:40
%S A244312 1,0,1,0,2,0,0,2,4,0,0,4,16,4,0,0,4,48,60,8,0,0,8,160,384,160,8,0,0,8,
%T A244312 368,1952,2176,520,16,0,0,16,1152,9648,18688,9648,1152,16,0,0,16,2432,
%U A244312 37008,132640,141680,45504,3568,32,0
%N A244312 Triangle read by rows: T(n,k) is the number of single loop solutions formed by n proper arches (connecting odd vertices to even vertices in the range 1 to 2n) above the x axis, k of which connect an odd vertex to a higher even vertex, with a rainbow of n arches below the x axis.
%C A244312 Sum of row n = (n-1)!.
%H A244312 Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, <a href="https://arxiv.org/abs/1807.01412">An asymptotic distribution theory for Eulerian recurrences with applications</a>, arXiv:1807.01412 [math.CO], 2018.
%F A244312 T(n,k)= (k+ floor((-1)^(n-1)/2))*T(n-1,k) + (n-k- floor((-1)^(n-1)/2))*T(n-1,k-1), n=>2, 1<=k<=n, T(1,1)=1, T(n,0)=0, T(n,n+1)=0.
%e A244312 Triangle T(n,k) begins:
%e A244312 n\k  1     2     3     4     5     6     7     8
%e A244312 1    1
%e A244312 2    0     1
%e A244312 3    0     2     0
%e A244312 4    0     2     4     0
%e A244312 5    0     4    16     4     0
%e A244312 6    0     4    48    60     8     0
%e A244312 7    0     8   160   384   160     8    0
%e A244312 8    0     8   368  1952  2176   520   16    0
%e A244312 T(4,3)=4 [top 14,23,56,78; bottom 18,27,36,45] [top 16,25,34,78; bottom 18,27,36,45] [top 12,34,58,67; bottom 18,27,36,45] [top 12,38,47,56; bottom 18,27,36,45]
%t A244312 T[1,1]:= 1; T[n_,0]:= 0; T[n_, n_+1] := 0; T[n_,k_]:= If[k == n+1, 0, (k + Floor[(-1)^(n-1)/2])*T[n-1, k] + (n-k -Floor[(-1)^(n-1)/2]) T[n-1, k - 1]]; Table[T[n, k], {n, 1, 15}, {k, 1, n}]//Flatten (* _G. C. Greubel_, Oct 10 2018 *)
%o A244312 (PARI) T(n,k)=if(n==1 && k==1, 1, if(k==0, 0, if( k==n+1, 0, (k+ floor((-1)^(n-1)/2))*T(n-1,k) + (n-k- floor((-1)^(n-1)/2))*T(n-1,k-1))));
%o A244312 for(n=1, 15, for(k=1,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 10 2018
%K A244312 nonn,tabl
%O A244312 1,5
%A A244312 _Roger Ford_, Jul 02 2014