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A054450 Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials). 9
1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 4, 4, 1, 1, 8, 8, 5, 5, 1, 1, 13, 12, 12, 6, 6, 1, 1, 21, 21, 17, 17, 7, 7, 1, 1, 34, 33, 33, 23, 23, 8, 8, 1, 1, 55, 55, 50, 50, 30, 30, 9, 9, 1, 1, 89, 88, 88, 73, 73, 38, 38, 10, 10, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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 Fib(z)/(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 first member of the family of Riordan-type matrices obtained from the unsigned convolution matrix A049310 by repeated application of the partial row sums procedure.

The column sequences are A000045(n+1) (Fibonacci), A052952, A054451 for m=0..2.

LINKS

Table of n, a(n) for n=0..65.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n, m)=sum(|A049310(n, k)|, k=m..n), (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) + |A049310(n, m)|, n >= m >= 0, a(n, m) := 0 if n<m.

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

The corresponding square array has T(n, k)=sum{j=0..floor(k/2), binomial(n+k-j, j)}. - Paul Barry, Oct 23 2004

EXAMPLE

{1}; {1,1}; {2,1,1}; {3,3,1,1};...

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

CROSSREFS

Cf. A049310, A000045. Row sums: A029907(n)= A054453(n, 0)

Sequence in context: A137896 A157219 A167040 * A174802 A238346 A053538

Adjacent sequences:  A054447 A054448 A054449 * A054451 A054452 A054453

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang, Apr 27 2000 and May 08 2000.

STATUS

approved

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Last modified April 20 08:37 EDT 2019. Contains 322306 sequences. (Running on oeis4.)