0,2

The g.f. for the sequence of column m of triangle A009766(n,m) (or Catalan A033184(n,n-m) diagonals) is N(2; m-1,x)*(x^m)/(1-x)^(m+1), m >= 1, with N(2; n,x) := sum(a(n,k)*x^k,k=0..n).

For k=0..1 the column sequences give A000108(n+1) (Catalan), -A002694. The row sums give A000012 (powers of 1) and (unsigned) A062992.

Another version of [1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [0, -1, -1, -1, -1, -1, -1, -1, ...] = 1; 1, 0; 2, -1, 0; 5, -6, 2, 0; 14, -28, 20, -5, 0; 42, -120, 135, -70, 14, 0; ... where DELTA is Deleham's operator defined in A084938.

The positive triangle has T(n,k)=binomial(2n+2,n-k)*binomial(n+k,k)/(n+1). - Paul Barry (pbarry(AT)wit.ie), May 11 2005

a(n, k) := [x^k]N(2; n, x) with N(2; n, x)=(N(2; n-1, x)-A000108(n)*(1-x)^(n+1))/x, N(2; 0, x) := 1.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=0, .., (n-2); a( n, k)= ((-1)^k)*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=(n-1) or n; else 0.

{1}; {2,-1}; {5,-6,2}; {14,-28,20,-5}; ...; N(2; 2,x)=5-6*x+2*x^2.

Sequence in context: A124575 A113345 A078123 this_sequence A118984 A073474 A067311

Adjacent sequences: A062988 A062989 A062990 this_sequence A062992 A062993 A062994

sign,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 12 2001