A109128 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 for 0

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

Offset: 0

Author

Reinhard Zumkeller, Jun 20 2005

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
		

Links

• Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A000035, A000295, A001861, A005408, A014473, A028387, A030662.

Cf. A134760, A281362, A327767.

Cf. A000325 (row sums).

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).

Programs

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

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 *)

Sage

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

Formula

T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 with T(n,0) = T(n,n) = 1.

Sum_{k=0..n} T(n, k) = A000325(n+1) (row sums).

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

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

From G. C. Greubel, Apr 06 2024: (Start)

T(n, n-k) = T(n, k).

T(2*n, n) = A134760(n).

T(2*n-1, n) = A030662(n), for n >= 1.

Sum_{k=0..n-1} T(n, k) = A000295(n+1), for n >= 1.

Sum_{k=0..n} (-1)^k*T(n, k) = 2*[n=0] - A000035(n+1).

Sum_{k=0..n-1} (-1)^k*T(n, k) = A327767(n), for n >= 1.

Sum_{k=0..floor(n/2)} T(n-k, k) = A281362(n).

Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A281362(n-1) - (1+(-1)^n)/2 for n >= 1.

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n), where b(n) is the repeating pattern {1,1,0,-2,-3,-1,2,2,-1,-3,-2,0} with b(n) = b(n-12). (End)

Extensions

Offset corrected by Reinhard Zumkeller, Apr 10 2012