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A244312 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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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

LINKS

Table of n, a(n) for n=1..55.

Hsien-Kuei Hwang, Hua-Huai Chern, Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 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.

EXAMPLE

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]

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 *)

PROG

(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

CROSSREFS

Sequence in context: A195581 A020474 A135589 * A158122 A028641 A141416

Adjacent sequences:  A244309 A244310 A244311 * A244313 A244314 A244315

KEYWORD

nonn,tabl,changed

AUTHOR

Roger Ford, Jul 02 2014

STATUS

approved

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Last modified October 21 08:42 EDT 2018. Contains 316405 sequences. (Running on oeis4.)