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A244312
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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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1
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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
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OFFSET
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1,5
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COMMENTS
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Sum of row n = (n-1)!.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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]
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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