A054448 Triangle of partial row sums of triangle A054446(n,m), n >= m >= 0.

1, 3, 1, 9, 4, 1, 26, 14, 5, 1, 73, 44, 20, 6, 1, 201, 131, 69, 27, 7, 1, 545, 376, 220, 102, 35, 8, 1, 1460, 1052, 665, 349, 144, 44, 9, 1, 3873, 2888, 1937, 1116, 528, 196, 54, 10, 1, 10191, 7813, 5490, 3402, 1788, 768, 259, 65, 11, 1, 26633, 20892, 15240, 10008

Offset

0,2

Comments

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 ((Pell(z))^2)/(Fib(z)*(1-x*z*Fib(z))) with Pell(x)=1/(1-2*x-x^2) = g.f. for A000129(n+1) (Pell numbers without 0) and 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 Fibonacci convolution matrix A037027 by repeated application of the partial row sums procedure.

Links

Table of n, a(n) for n=0..58.

Formula

a(n, m)=sum(A054446(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)*A037027(n-j, 0), j=m..n) + A054446(n, m), n >= m >= 0, a(n, m) := 0 if n<m.

G.f. for column m: (((Pell(x))^2)/Fib(x))*(x*Fib(x))^m, m >= 0, with Fib(x) = g.f. A000045(n+1) and Pell(x) = g.f. A000129(n+1).

Example

{1}; {3,1}; {9,4,1}; {26,14,5,1};...
Fourth row polynomial (n=3): p(3,x)= 26+14*x+5*x^2+x^3

Crossrefs

Cf. A037027, A000045, A000129. Row sums: A054449.

Sequence in context: A368568 A370150 A162852 * A106516 A140071 A285280

Adjacent sequences: A054445 A054446 A054447 * A054449 A054450 A054451

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang, Apr 27 2000 and May 08 2000.

Status

approved