0,8

This sequence provides the general solution to the recurrence a(n)=a(n-1)+k(k+1)a(n-2), a(0)=a(1)=1. The solution is (1,1,k^2+k+1,2k^2+2k+1, .....) whose coefficients can be read from the rows of the triangle. The row sums of the triangle are given by the case k=1. These are the Jacobsthal numbers, A001045. Viewed as a square array, its first row is (1,0,1,0,1,...) with E.g.f. cosh(x), G.f. 1/(1-x^2) and subsequent rows are successive partial sums given by 1/((1-x)^n)(1-x^2)). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003

G.f.: 1/(1-x-x*y-y^2)

As a square array read by antidiagonals, this is T(n, k)=sum{i=0..n, (-1)^(n-i)C(i+k, k)} - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003

T(2*n,n)=A026641(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 08 2007

Triangle begins:

1;

1,0;

1,1,1;

1,2,2,0;

1,3,4,2,1;

1,4,7,6,3,0;

1,5,11,13,9,3,1;

1,6,16,24,22,12,4,0;

1,7,22,40,46,34,16,4,1;

1,8,29,62,86,80,50,20,5,0;

1,9,37,91,148,166,130,70,25,5,1;

1,10,46,128,239,314,296,200,95,30,6,0;

...

read transforms; 1/(1-x-x*y-y^2); SERIES2(%, x, y, 12); SERIES2TOLIST(%, x, y, 12);

See A059260 for an explicit formula.

Diagonals of this triangle are given by A006498

Similar to the triangles A035317, A080242, A108561, A112555.

Sequence in context: A104245 A167637 A109754 this_sequence A124394 A086460 A136431

Adjacent sequences: A059256 A059257 A059258 this_sequence A059260 A059261 A059262

nonn,tabl

N. J. A. Sloane (njas(AT)research.att.com), Jan 23 2001