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Triangle read by rows: T(n,k) is the number of single loop solutions formed by n proper arches (connecting an odd and even vertice from 1 to 2n) above the x axis, k arches above the x axis connecting an odd vertice to a higher even vertice and a rainbow of n arches below the x axis.
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%I #26 Oct 10 2018 10:49:32

%S 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 368,1952,2176,520,16,0,0,16,1152,9648,18688,9648,1152,16,0,0,16,2432,

%U 37008,132640,141680,45504,3568,32,0

%N Triangle read by rows: T(n,k) is the number of single loop solutions formed by n proper arches (connecting an odd and even vertice from 1 to 2n) above the x axis, k arches above the x axis connecting an odd vertice to a higher even vertice and a rainbow of n arches below the x axis.

%C Sum of row n = (n-1)!.

%H Hsien-Kuei Hwang, Hua-Huai Chern, 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 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 Triangle T(n,k) begins:

%e n\k 1 2 3 4 5 6 7 8

%e 1 1

%e 2 0 1

%e 3 0 2 0

%e 4 0 2 4 0

%e 5 0 4 16 4 0

%e 6 0 4 48 60 8 0

%e 7 0 8 160 384 160 8 0

%e 8 0 8 368 1952 2176 520 16 0

%e 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 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 (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 for(n=1, 15, for(k=1,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 10 2018

%K nonn,tabl

%O 1,5

%A _Roger Ford_, Jul 02 2014