A131268 Triangle read by rows: T(n,k) = 2*binomial(n-floor((k+1)/2),floor(k/2)) - 1, 0<=k<=n.

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 1, 7, 5, 5, 1, 1, 1, 9, 7, 11, 5, 1, 1, 1, 11, 9, 19, 11, 7, 1, 1, 1, 13, 11, 29, 19, 19, 7, 1, 1, 1, 15, 13, 41, 29, 39, 19, 9, 1, 1, 1, 17, 15, 55, 41, 69, 39, 29, 9, 1, 1, 1, 19, 17, 71, 55, 111, 69, 69, 29, 11, 1, 1, 1, 21, 19, 89, 71, 167, 111, 139, 69, 41, 11, 1

Offset: 0

Author

Gary W. Adamson, Jun 23 2007

Keywords

• nonn
• tabl

Comments

Row sums are in A131269. Reversal = triangle A131270.

Examples

			Triangle begins:
1;
1, 1;
1, 1,  1;
1, 1,  3,  1;
1, 1,  5,  3,  1;
1, 1,  7,  5,  5,  1;
1, 1,  9,  7, 11,  5,   1;
1, 1, 11,  9, 19, 11,   7,   1;
1, 1, 13, 11, 29, 19,  19,   7,   1;
1, 1, 15, 13, 41, 29,  39,  19,   9,  1;
1, 1, 17, 15, 55, 41,  69,  39,  29,  9,  1;
1, 1, 19, 17, 71, 55, 111,  69,  69, 29, 11,  1;
1, 1, 21, 19, 89, 71, 167, 111, 139, 69, 41, 11, 1;
...
		

Links

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

Crossrefs

Cf. A065941, A000012, A131269, A131270.

Programs

Magma

[2*Binomial(n-Floor((k+1)/2), Floor(k/2))-1: k in [0..n], n in [0..14]]; // Bruno Berselli, May 03 2012

Maple

T := proc (n, k) options operator, arrow; 2*binomial(n-floor((1/2)*k+1/2), floor((1/2)*k))-1 end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form. - Emeric Deutsch, Jul 15 2007

Mathematica

Table[2*Binomial[n -Floor[(k+1)/2], Floor[k/2]] -1, {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 10 2019 *)

PARI

T(n,k) = 2*binomial(n- (k+1)\2, k\2) -1; \\ G. C. Greubel, Jul 10 2019

Sage

[[2*binomial(n -floor((k+1)/2), floor(k/2)) -1 for k in (0..n)] for n in (0..14)] # G. C. Greubel, Jul 10 2019

Formula

Equals 2*A065941 - A000012, where A065941 = Pascal's triangle with repeated columns; and A000012 = (1; 1,1; 1,1,1;...) as an infinite lower triangular matrix.

Extensions

More terms from Emeric Deutsch, Jul 15 2007