A098574 -id:A098574 - cp's OEIS frontend

A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 11, 15, 15, 11, 5, 1, 16, 26, 30, 26, 16, 6, 1, 22, 42, 56, 56, 42, 22, 7, 1, 29, 64, 98, 112, 98, 64, 29, 8, 1, 37, 93, 162, 210, 210, 162, 93, 37, 9, 1, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10, 1, 56, 176, 385, 627, 792, 792, 627

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 05 2006

Binomial transform of the infinite tridiagonal matrix with main diagonal, (1,1,1,...), subdiagonal, (0,0,0,...) and subsubdiagonal, (1,1,1,...). Sum of entries in row n = 2^(n+1) - n - 1 = A000325(n+1).

Riordan array ((1-2x+2x^2)/(1-x)^3, x/(1-x)). - Paul Barry, Apr 08 2011

			Row 3 = (4, 4, 3, 1), then row 4 = (7, 8, 7, 4, 1).
First few rows of the triangle are
   1;
   1,  1;
   2,  2,  1;
   4,  4,  3,  1;
   7,  8,  7,  4,  1;
  11, 15, 15, 11,  5,  1;
  16, 26, 30, 26, 16,  6,  1;
  ...
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins
   1, 1;
   1, 1, 1;
   0, 0, 1, 1;
  -1, 0, 0, 1, 1;
   0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 1, 1;
  -1, 0, 0, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
(End)

Cf. A098574, A000125, A055795, A027660, A055796, A002663.

Cf. A000325.

T:=(n,k)->binomial(n,k)+binomial(n,k+2): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Flatten[Table[Binomial[n,k]+Binomial[n,k+2],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Jun 12 2015 *)

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 12 2014

Edited by N. J. A. Sloane, Nov 29 2006

A217839 T(n,k)=Number of n element 0..1 arrays with each element the minimum of k adjacent elements of a random 0..1 array of n+k-1 elements.

Original entry on oeis.org

2, 2, 4, 2, 4, 8, 2, 4, 7, 16, 2, 4, 7, 12, 32, 2, 4, 7, 11, 21, 64, 2, 4, 7, 11, 17, 37, 128, 2, 4, 7, 11, 16, 27, 65, 256, 2, 4, 7, 11, 16, 23, 44, 114, 512, 2, 4, 7, 11, 16, 22, 34, 72, 200, 1024, 2, 4, 7, 11, 16, 22, 30, 52, 117, 351, 2048, 2, 4, 7, 11, 16, 22, 29, 42, 81, 189, 616

Offset: 1

R. H. Hardin Oct 12 2012

Table starts
.....2....2....2....2...2...2...2...2...2...2...2...2...2...2
.....4....4....4....4...4...4...4...4...4...4...4...4...4...4
.....8....7....7....7...7...7...7...7...7...7...7...7...7...7
....16...12...11...11..11..11..11..11..11..11..11..11..11..11
....32...21...17...16..16..16..16..16..16..16..16..16..16..16
....64...37...27...23..22..22..22..22..22..22..22..22..22..22
...128...65...44...34..30..29..29..29..29..29..29..29..29..29
...256..114...72...52..42..38..37..37..37..37..37..37..37..37
...512..200..117...81..61..51..47..46..46..46..46..46..46..46
..1024..351..189..126..91..71..61..57..56..56..56..56..56..56
..2048..616..305..194.137.102..82..72..68..67..67..67..67..67
..4096.1081..493..296.205.149.114..94..84..80..79..79..79..79
..8192.1897..798..450.303.218.162.127.107..97..93..92..92..92
.16384.3329.1292..685.443.316.232.176.141.121.111.107.106.106
.32768.5842.2091.1046.644.452.331.247.191.156.136.126.122.121

			Some solutions for n=8 k=4
..0....0....1....0....1....0....1....1....0....1....0....0....0....0....0....0
..1....1....1....1....0....0....1....1....1....0....0....1....0....1....0....0
..1....1....1....1....0....1....0....1....1....0....1....0....1....0....0....0
..1....1....0....0....0....0....0....0....0....0....1....0....1....0....1....0
..1....1....0....0....0....0....0....0....0....0....1....0....1....0....1....1
..1....1....0....0....0....0....0....0....0....1....1....0....1....0....1....0
..1....0....0....0....0....0....0....0....0....1....1....0....0....1....1....0
..0....0....1....0....1....0....1....0....1....1....1....1....0....1....0....0

R. H. Hardin, Table of n, a(n) for n = 1..435

Column 2 is A005251(n+3)
Column 3 is A005252(n+3)
Column 4 is A005253(n+3)
Column 5 is A005689(n+6)
Column 6 is A098574(n+6)
Diagonal is A000124

Empirical for columns 1-7: a(n) = 2*a(n-1) -a(n-2) +a(n-k-1)

cp's OEIS Frontend

A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).

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