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 A026736 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for n >= 0; for n >= 2 and 1 <= k <= n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if n is even and k=(n-2)/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k). 31
 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 4, 1, 1, 6, 11, 10, 5, 1, 1, 7, 22, 21, 15, 6, 1, 1, 8, 29, 43, 36, 21, 7, 1, 1, 9, 37, 94, 79, 57, 28, 8, 1, 1, 10, 46, 131, 173, 136, 85, 36, 9, 1, 1, 11, 56, 177, 398, 309, 221, 121, 45, 10, 1, 1, 12, 67, 233, 575, 707, 530, 342, 166, 55, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(n, k) is the number of paths from (0, 0) to (n-k, 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, i+2)-to-(i+1, i+3) for i >= 0. LINKS G. C. Greubel, Rows n = 0..100 of triangle, flattened EXAMPLE Triangle begins 1; 1, 1; 1, 2, 1; 1, 3, 3, 1; 1, 5, 6, 4, 1; 1, 6, 11, 10, 5, 1; 1, 7, 22, 21, 15, 6, 1; 1, 8, 29, 43, 36, 21, 7, 1; 1, 9, 37, 94, 79, 57, 28, 8, 1; 1, 10, 46, 131, 173, 136, 85, 36, 9, 1; 1, 11, 56, 177, 398, 309, 221, 121, 45, 10, 1; 1, 12, 67, 233, 575, 707, 530, 342, 166, 55, 11, 1; ... MATHEMATICA T[_, 0] = T[n_, n_] = 1; T[n_, k_] := T[n, k] = If[EvenQ[n] && 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, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 22 2018 *) PROG (PARI) T(n, k) = if(k==n || k==0, 1, if((n%2)==0 && 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=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 16 2019 (Sage) def T(n, k): if (k==0 or k==n): return 1 elif (mod(n, 2)==0 and 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 (0..n)] for n in (0..12)] # G. C. Greubel, Jul 16 2019 (GAP) T:= function(n, k) if k=0 or k=n then return 1; elif (n mod 2)=0 and 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([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Jul 16 2019 CROSSREFS Row sums give A026743. T(2n,n) gives A026737(n) or A111279(n+1). Sequence in context: A138201 A220614 A154221 * A230859 A213086 A050446 Adjacent sequences: A026733 A026734 A026735 * A026737 A026738 A026739 KEYWORD nonn,tabl,walk AUTHOR Clark Kimberling EXTENSIONS Offset corrected by Alois P. Heinz, Jul 23 2018 STATUS approved

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Last modified September 18 08:46 EDT 2024. Contains 375999 sequences. (Running on oeis4.)