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A027948 Triangular array T read by rows: T(n,k) = t(n,2k+1) for 0 <= k <= n, T(n,n)=1, t given by A027926, n >= 0. 13
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 7, 4, 1, 1, 3, 8, 14, 5, 1, 1, 3, 8, 20, 25, 6, 1, 1, 3, 8, 21, 46, 41, 7, 1, 1, 3, 8, 21, 54, 97, 63, 8, 1, 1, 3, 8, 21, 55, 133, 189, 92, 9, 1, 1, 3, 8, 21, 55, 143, 309, 344, 129, 10, 1, 1, 3, 8, 21, 55, 144, 364, 674, 591, 175, 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) = Sum_{j=0..n-k} binomial(n-j, 2*(n-k-j) -1) with T(n,n)=1 in the region n >= 0, 0 <= k <= n. - G. C. Greubel, Sep 29 2019

EXAMPLE

Triangle begins with:

  1;

  1, 1;

  1, 2, 1;

  1, 3, 3,  1;

  1, 3, 7,  4,  1;

  1, 3, 8, 14,  5,  1;

  1, 3, 8, 20, 25,  6, 1;

  1, 3, 8, 21, 46, 41, 7, 1; ...

MAPLE

T:= proc(n, k)

      if k=n then 1

      else add(binomial(n-j, 2*(n-k-j)-1), j=0..n-k)

      fi

    end:

seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Sep 29 2019

MATHEMATICA

T[n_, k_]:= If[k==n, 1, Sum[Binomial[n-j, 2*(n-k-j)-1], {j, 0, n-k}]];

Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 29 2019 *)

PROG

(PARI) T(n, k) = if(k==n, 1, sum(j=0, n-k, binomial(n-j, 2*(n-k-j)-1)) );

for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Sep 29 2019

(MAGMA) T:= func< n, k | k eq n select 1 else &+[Binomial(n-j, 2*(n-k-j) -1): j in [0..n-k]] >;

[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2019

(Sage)

def T(n, k):

    if (k==n): return 1

    else: return sum(binomial(n-j, 2*(n-k-j)-1) for j in (0..n-k))

[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 29 2019

(GAP)

T:= function(n, k)

    if k=n then return 1;

    else return Sum([0..n-k], j-> Binomial(n-j, 2*(n-k-j)-1) );

    fi;

  end;

Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Sep 29 2019

CROSSREFS

The row sums of this (slightly extended) bisection of the "Fibonacci array" A027926 are powers of 2, see A027935 for the other bisection.

Sequence in context: A174374 A242641 A347187 * A095141 A177974 A095140

Adjacent sequences:  A027945 A027946 A027947 * A027949 A027950 A027951

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

EXTENSIONS

Name edited by G. C. Greubel, Sep 29 2019

STATUS

approved

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Last modified December 3 17:23 EST 2021. Contains 349467 sequences. (Running on oeis4.)