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 A026758 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 odd and 1 <= k <= (n-1)/2, else T(n,k) = T(n-1,k-1) + T(n-1,k). 30
 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 7, 4, 1, 1, 7, 16, 11, 5, 1, 1, 8, 23, 27, 16, 6, 1, 1, 10, 38, 66, 43, 22, 7, 1, 1, 11, 48, 104, 109, 65, 29, 8, 1, 1, 13, 69, 190, 279, 174, 94, 37, 9, 1, 1, 14, 82, 259, 469, 453, 268, 131, 46, 10, 1, 1, 16, 109, 410, 918, 1201, 721, 399, 177, 56, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS G. C. Greubel, Rows n = 0..100 of triangle, flattened FORMULA T(n, k) = 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, 2h+i+1)-to-(i+1, 2h+i+2) for i >= 0, h>=0. EXAMPLE Triangle begins as:   1;   1,  1;   1,  2,  1;   1,  4,  3,  1;   1,  5,  7,  4,  1;   1,  7, 16, 11,  5,  1;   1,  8, 23, 27, 16,  6, 1;   1, 10, 38, 66, 43, 22, 7, 1; MAPLE T:= proc(n, k) option remember;    if k=0 or k = n then 1;    elif type(n, 'odd') and k <= (n-1)/2 then         procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;    else        procname(n-1, k-1)+procname(n-1, k) ;    end if ; end proc; seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Oct 29 2019 MATHEMATICA T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[OddQ[n] && k<=(n-1)/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 (* G. C. Greubel, Oct 29 2019 *) PROG (PARI) T(n, k) = if(k==0 || k==n, 1, if(n%2==1 && k<=(n-1)/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, Oct 29 2019 (Sage) @CachedFunction def T(n, k):     if (k==0 or k==n): return 1     elif (mod(n, 2)==1 and k<=(n-1)/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, Oct 29 2019 (GAP) T:= function(n, k)     if k=0 or k=n then return 1;     elif (n mod 2)=1 and k List([0..n], k-> T(n, k) ))); # G. C. Greubel, Oct 29 2019 CROSSREFS Cf. A026765 (row sums). Sequence in context: A229118 A320796 A026725 * A130523 A034363 A026769 Adjacent sequences:  A026755 A026756 A026757 * A026759 A026760 A026761 KEYWORD nonn,tabl AUTHOR EXTENSIONS Offset corrected by Sean A. Irvine, Oct 25 2019 More terms added by G. C. Greubel, Oct 29 2019 STATUS approved

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Last modified September 28 19:16 EDT 2021. Contains 347717 sequences. (Running on oeis4.)