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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<Int(n/2)+1 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, 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,changed

AUTHOR

Clark Kimberling

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 November 11 18:50 EST 2019. Contains 329031 sequences. (Running on oeis4.)