A080233 - OEIS

%I #22 Mar 25 2025 11:59:42

%S 1,1,0,1,1,-1,1,2,0,-2,1,3,2,-2,-3,1,4,5,0,-5,-4,1,5,9,5,-5,-9,-5,1,6,

%T 14,14,0,-14,-14,-6,1,7,20,28,14,-14,-28,-20,-7,1,8,27,48,42,0,-42,

%U -48,-27,-8,1,9,35,75,90,42,-42,-90,-75,-35,-9

%N Triangle T(n,k) obtained by taking differences of consecutive pairs of row elements of Pascal's triangle A007318.

%C Row sums are 1,1,1,1,1,1 with g.f. 1/(1-x). Can also be obtained from triangle A080232 by taking sums of pairs of consecutive row elements.

%C Mirror image of triangle in A156644. - _Philippe Deléham_, Feb 14 2009

%F T(n, k) = if(k>n, 0, binomial(n, k)-binomial(n, k-1)).

%e Triangle begins as:

%e   1;

%e   1, 0;

%e   1, 1, -1;

%e   1, 2,  0, -2;

%e   1, 3,  2, -2, -3;

%e   1, 4,  5,  0, -5,  -4;

%e   1, 5,  9,  5, -5,  -9,  -5;

%e   1, 6, 14, 14,  0, -14, -14,  -6;

%e   1, 7, 20, 28, 14, -14, -28, -20,  -7;

%e   1, 8, 27, 48, 42,   0, -42, -48, -27,  -8;

%e   1, 9, 35, 75, 90,  42, -42, -90, -75, -35, -9;

%e   ...

%t Table[Binomial[n, k] - Binomial[n, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Nov 24 2016 *)

%o (PARI) {T(n, k) = if( n<0 || k>n, 0, binomial(n, k) - binomial(n, k-1))}; /* _Michael Somos_, Nov 25 2016 */

%Y Row sums give A000012.

%Y Cf. A007318, A080232.

%K easy,sign,tabl

%O 0,8

%A _Paul Barry_, Feb 10 2003