0,2

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((1-z^2)*(Fib(z))^2)/(1-x*z/(1-z^2)) Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).

This is the second member of the family of Riordan-type matrices obtained from the unsigned convolution matrix A049310(n,m) by repeated application of the partial row sums procedure.

The column sequences are A029907, A001629, A054454 for m=0..2.

a(n, m)=sum(A054450(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n<m, (sequence of partial row sums in column m).

Column m recursion: a(n, m)= sum(a(j-1, m)*|A049310(n-j, 0)|, j=m..n) + A054450(n, m), n >= m >= 0, a(n, m) := 0 if n<m.

G.f. for column m: ((1-x^2)*(Fib(x))^2)*(x/(1-x^2))^m, m >= 0, with Fib(x) G.f. for A000045(n+1).

{1}; {2,1}; {4,2,1}; {8,5,2,1};...

Fourth row polynomial (n=3): p(3,x)= 8+5*x+2*x^2+x^3

Cf. A049310, A054450, A000045, A029907, A001629. Row sums: A054455(n).

Sequence in context: A130321 A101508 A106471 this_sequence A109433 A123490 A060637

Adjacent sequences: A054450 A054451 A054452 this_sequence A054454 A054455 A054456

easy,nonn,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 27 2000 and May 08 2000.