0,1

This is the third member, q=3, in the family of (1,q) Pascal triangles: A007318 (Pascal (q=1), A029635 (q=2) (but with a(0,0)=2, not 1).

This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column nr. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n) is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(3-2*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(3-2*z)/(1-(1+x)*z).

The SW-NE diagonals give sum(a(n-1-k,k),k=0..ceiling((n-1)/2)) = A000285(n-2), n>=2, with n=1 value 3. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs.

W. Lang, First 10 rows.

Recursion: a(n, m)=0 if m>n, a(0, 0)= 3; a(n, 0)=1 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).

G.f. column m (without leading zeros): (3-2*x)/(1-x)^(m+1), m>=0.

[3];[1,3];[1,4,3];[1,5,7,3];[1,6,12,10,3];...

Row sums: A000079(n+1), n>=1, 3 if n=0. Alternating row sums are [3, -2, followed by 0's].

Column sequences (without leading zeros) give for m=1..9 with n>=0: A000027(n+3), A055998(n+1), A006503(n+1), A095661, A000574, A095662-5.

Adjacent sequences: A095657 A095658 A095659 this_sequence A095661 A095662 A095663

Sequence in context: A057024 A023892 A085417 this_sequence A035648 A053575 A103790

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), May 21 2004