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.
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, 368, 1952, 2176, 520, 16, 0, 0, 16, 1152, 9648, 18688, 9648, 1152, 16, 0, 0, 16, 2432, 37008, 132640, 141680, 45504, 3568, 32, 0
Offset: 1
Examples
Triangle T(n,k) begins: n\k 1 2 3 4 5 6 7 8 1 1 2 0 1 3 0 2 0 4 0 2 4 0 5 0 4 16 4 0 6 0 4 48 60 8 0 7 0 8 160 384 160 8 0 8 0 8 368 1952 2176 520 16 0 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]
Links
- Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018.
Programs
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Mathematica
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 *) -
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)))); for(n=1, 15, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 10 2018
Formula
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.
Comments