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A026725 Triangular array, T, read by rows: T(n,0) = T(n,n) = 1, 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, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k). 26
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 7, 4, 1, 1, 6, 16, 11, 5, 1, 1, 7, 22, 27, 16, 6, 1, 1, 8, 29, 65, 43, 22, 7, 1, 1, 9, 37, 94, 108, 65, 29, 8, 1, 1, 10, 46, 131, 267, 173, 94, 37, 9, 1, 1, 11, 56, 177, 398, 440, 267, 131, 46, 10, 1, 1, 12, 67, 233 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

T(n+2,n) = A134869(n+1). - Philippe Deléham, Feb 01 2014

LINKS

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Rob Arthan, Comments on A026674, A026725, A026670

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, i+1)-to-(i+1, i+2) for i >= 0.

Comment from Rick L. Shepherd, Aug 05 2002: Probably this should be changed to "and edges (i+1, i)-to-(i+2, i+1) for i >= 0."

EXAMPLE

Triangle begins:

1

1  1

1  2  1

1  4  3   1

1  5  7   4   1

1  6 16  11   5    1

1  7 22  27  16    6   1

1  8 29  65  43   22   7   1

1  9 37  94 108   65  29   8   1

1 10 46 131 267  173  94  37   9  1

1 11 56 177 398  440 267 131  46 10  1

1 12 67 233 575 1105 707 398 177 56 11 1

... - Philippe Deléham, Feb 01 2014

MATHEMATICA

T[n_, k_]:= T[n, k] = If[k==1 || k==n, 1, If[EvenQ[n] && k==n/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, 15}, {k, n}]//Flatten (* G. C. Greubel, Jul 16 2019 *)

PROG

(PARI) T(n, k) = if(k==n || k==1, 1, if((n%2)==0 && k==n/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 16 2019

(Sage)

def T(n, k):

    if (k==1 or k==n): return 1

    elif (mod(n, 2)==0 and k==n/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..15)] # G. C. Greubel, Jul 16 2019

(GAP)

T:= function(n, k)

    if k=1 or k=n then return 1;

    elif (n mod 2)=0 and k=Int(n/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..15], n-> List([1..n], k-> T(n, k) ))); # G. C. Greubel, Jul 16 2019

CROSSREFS

Cf. A026674.

Sequence in context: A323182 A229118 A320796 * A026758 A130523 A034363

Adjacent sequences:  A026722 A026723 A026724 * A026726 A026727 A026728

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

EXTENSIONS

Title corrected by G. C. Greubel, Jul 16 2019

STATUS

approved

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Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)