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A132731 Triangle T(n,k) = 2 * binomial(n,k) - 2 with T(n,0) = T(n,n) = 1, read by rows. 4
1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 8, 18, 18, 8, 1, 1, 10, 28, 38, 28, 10, 1, 1, 12, 40, 68, 68, 40, 12, 1, 1, 14, 54, 110, 138, 110, 54, 14, 1, 1, 16, 70, 166, 250, 250, 166, 70, 16, 1, 1, 18, 88, 238, 418, 502, 418, 238, 88, 18, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

FORMULA

T(n, k) = 2*A007318 + A103451 - 2*A000012, an infinite lower triangular matrix.

From G. C. Greubel, Feb 14 2021: (Start)

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

T(n, k) = 2*A132044(n, k) with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n, k) = 2^(n+1) - 2*n - [n=0] = A132732(n). (End)

EXAMPLE

First few rows of the triangle are:

  1;

  1,  1;

  1,  2,  1;

  1,  4,  4,  1;

  1,  6, 10,  6,  1;

  1,  8, 18, 18,  8,  1;

  1, 10, 28, 38, 28, 10,  1;

  1, 12, 40, 68, 68, 40, 12, 1;

  ...

MATHEMATICA

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

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)

PROG

(PARI) t(n, k) =  2*binomial(n, k) + ((k==0) || (k==n)) - 2*(k<=n); \\ Michel Marcus, Feb 12 2014

(Sage)

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

flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021

(Magma)

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

[T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel, Feb 14 2021

CROSSREFS

Cf. A000012, A007318, A103451, A132044, A132732 (row sums).

Sequence in context: A283796 A156580 A157528 * A128966 A055907 A259698

Adjacent sequences:  A132728 A132729 A132730 * A132732 A132733 A132734

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Aug 26 2007

EXTENSIONS

Corrected by Jeremy Gardiner, Feb 02 2014

More terms from Michel Marcus, Feb 12 2014

STATUS

approved

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Last modified July 30 12:39 EDT 2021. Contains 346359 sequences. (Running on oeis4.)