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A132729 Triangle T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1, read by rows. 3
1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 9, 5, 1, 1, 7, 17, 17, 7, 1, 1, 9, 27, 37, 27, 9, 1, 1, 11, 39, 67, 67, 39, 11, 1, 1, 13, 53, 109, 137, 109, 53, 13, 1, 1, 15, 69, 165, 249, 249, 165, 69, 15, 1, 1, 17, 87, 237, 417, 501, 417, 237, 87, 17, 1, 1, 19, 107, 327, 657, 921, 921, 657, 327, 107, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

FORMULA

T(n, k) = 2*A132044(n, k) - 1.

From G. C. Greubel, Feb 13 2021: (Start)

T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n, k) = 2^(n+1) - 3*n + 1 - 2*[n=0] = A132730(n). (End)

EXAMPLE

First few rows of the triangle are:

  1;

  1,  1;

  1,  1,  1;

  1,  3,  3,  1;

  1,  5,  9,  5,  1;

  1,  7, 17, 17,  7,  1;

  1,  9, 27, 37, 26,  9,  1;

  1, 11, 39, 67, 67, 39, 11, 1;

MATHEMATICA

T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] - 3];

Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 13 2021 *)

PROG

(Sage)

def T(n, k): return 1 if (k==0 or k==n) else 2*binomial(n, k) - 3

flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

(Magma)

T:= func< n, k | k eq 0 or k eq n select 1 else 2*Binomial(n, k) - 3 >;

[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

CROSSREFS

Cf. A007318, A132044, A132730.

Sequence in context: A056611 A177407 A135669 * A196493 A251634 A196989

Adjacent sequences:  A132726 A132727 A132728 * A132730 A132731 A132732

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Aug 26 2007

EXTENSIONS

More terms added by G. C. Greubel, Feb 13 2021

STATUS

approved

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Last modified June 19 00:37 EDT 2021. Contains 345125 sequences. (Running on oeis4.)