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A026703
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Triangular array T read by rows: T(n,1) = T(n,n) = 1, T(n,k) = T(n-1, k-1) + T(n-2,k-1) + T(n-1,k) if k=(n/2) or k=((n+1)/2), otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).
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16
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1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 6, 13, 6, 1, 1, 7, 24, 24, 7, 1, 1, 8, 31, 61, 31, 8, 1, 1, 9, 39, 116, 116, 39, 9, 1, 1, 10, 48, 155, 293, 155, 48, 10, 1, 1, 11, 58, 203, 564, 564, 203, 58, 11, 1, 1, 12, 69, 261, 767, 1421, 767, 261, 69, 12, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,5
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LINKS
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FORMULA
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T(n, k) = number of paths from (0, 0) to (n-k, k) in the 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) if |i-j|<=1.
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EXAMPLE
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Triangle begins:
1,
1, 1,
1, 3, 1,
1, 5, 5, 1,
1, 6, 13, 6, 1,
1, 7, 24, 24, 7, 1,
1, 8, 31, 61, 31, 8, 1,
1, 9, 39, 116, 116, 39, 9, 1,
...
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MATHEMATICA
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T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, If[k==n/2 || k==(n+1)/2 || k== (n+2)/2, T[n-1, k-1] +T[n-2, k-1] +T[n-1, k], T[n-1, k-1] +T[n-1, k]]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 17 2019 *)
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PROG
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(PARI) T(n, k) = if(k==1 || k==n, 1, if(k==n/2 || k==(n+1)/2 || k==(n+2)/2 , T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 17 2019
(Sage)
def T(n, k):
if (k==1 or k==n): return 1
elif (k==n/2 or k==(n+1)/2 or k==(n+2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 17 2019
(GAP)
T:= function(n, k)
if (k=1 or k=n) then return 1;
elif (k=Int(n/2) and k=Int((n+1)/2) or k=Int((n+2)/2)) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
else return T(n-1, k-1) + T(n-1, k);
fi;
end;
Flat(List([1..12], n-> List([1..n], k-> T(n, k) ))); # G. C. Greubel, Jul 17 2019
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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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