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A109128 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 for 0<k<n, T(n,0) = T(n,n) = 1. 22
1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 11, 7, 1, 1, 9, 19, 19, 9, 1, 1, 11, 29, 39, 29, 11, 1, 1, 13, 41, 69, 69, 41, 13, 1, 1, 15, 55, 111, 139, 111, 55, 15, 1, 1, 17, 71, 167, 251, 251, 167, 71, 17, 1, 1, 19, 89, 239, 419, 503, 419, 239, 89, 19, 1, 1, 21, 109, 329, 659, 923, 923, 659, 329, 109, 21, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Eigensequence of the triangle = A001861. - Gary W. Adamson, Apr 17 2009

T(n,k) = A014473(n,k) + A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 10 2012

LINKS

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

T(n, k) = 2*binomial(n,k) - 1. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Sep 30 2007

T(n, 1) = 2*n - 1 = A005408(n+1) for n>0.

T(n, 2) = n^2 + n - 1 = A028387(n-2) for n>1.

T(n, k) = Sum_{j=0..n-k} C(n-k,j)*C(k,j)*(2 - 0^j) for k <= n - Paul Barry, Apr 27 2006

EXAMPLE

Triangle begins as:

  1;

  1   1;

  1   3   1;

  1   5   5   1;

  1   7  11   7   1;

  1   9  19  19   9   1;

  1  11  29  39  29  11   1;

  1  13  41  69  69  41  13   1;

  1  15  55 111 139 111  55  15   1;

  1  17  71 167 251 251 167  71  17   1;

  1  19  89 239 419 503 419 239  89  19   1;

MAPLE

A109128 := proc(n, k)

    2*binomial(n, k)-1 ;

end proc: # R. J. Mathar, Jul 12 2016

MATHEMATICA

Table[2*Binomial[n, k] -1, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

PROG

(Haskell)

a109128 n k = a109128_tabl !! n !! k

a109128_row n = a109128_tabl !! n

a109128_tabl = iterate (\row -> zipWith (+)

   ([0] ++ row) (1 : (map (+ 1) $ tail row) ++ [0])) [1]

-- Reinhard Zumkeller, Apr 10 2012

(MAGMA) [2*Binomial(n, k) -1: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 12 2020

(Sage) [[2*binomial(n, k) -1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020

CROSSREFS

Cf. A000325 (row sums), A001861.

Sequence m*binomial(n,k) - (m-1): A007318 (m=1), this sequence (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), A131067 (m=7), A168625 (m=8).

Sequence in context: A134398 A026615 A026681 * A113245 A103450 A128254

Adjacent sequences:  A109125 A109126 A109127 * A109129 A109130 A109131

KEYWORD

nonn,tabl

AUTHOR

Reinhard Zumkeller, Jun 20 2005

EXTENSIONS

Offset corrected by Reinhard Zumkeller, Apr 10 2012

STATUS

approved

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Last modified April 7 04:20 EDT 2020. Contains 333292 sequences. (Running on oeis4.)