

A026692


Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n1, T(n,k)=T(n1,k1)+T(n2,k1)+T(n1,k) if k or nk is of form 2h for h=1,2,...,[ n/4 ], else T(n,k)=T(n1,k1)+T(n1,k).


15



1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 15, 15, 5, 1, 1, 6, 24, 30, 24, 6, 1, 1, 7, 35, 54, 54, 35, 7, 1, 1, 8, 48, 89, 138, 89, 48, 8, 1, 1, 9, 63, 137, 281, 281, 137, 63, 9, 1, 1, 10, 80, 200, 507, 562, 507, 200, 80, 10, 1, 1, 11, 99, 280, 844
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OFFSET

1,5


LINKS



FORMULA

T(n, k) = number of paths from (0, 0) to (nk, k) in directed graph having vertices (i, j) and edges (i, j)to(i+1, j) and (i, j)to(i, j+1) for i, j >= 0 and edges (i, j)to(i+1, j+1) for i odd and j >= i and for j odd and i >= j.


EXAMPLE

Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 8, 4, 1;
1, 5, 15, 15, 5, 1;
...


MATHEMATICA

T[_, 0] = 1; T[n_, n_] = 1; T[n_, k_] /; EvenQ[k] && 1 <= k/2 <= Floor[n/4]  EvenQ[nk] && 1 <= (nk)/2 <= Floor[n/4] := T[n, k] = T[n1, k1] + T[n2, k1] + T[n1, k]; T[n_, k_] := T[n, k] = T[n1, k1] + T[n1, k];


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



