

A054450


Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev Spolynomials).


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
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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 Riordangroup. The G.f. for the row polynomials p(n,x) (increasing powers of x) is Fib(z)/(1x*z/(1z^2)) Fib(x)=1/(1xx^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).
This is the first member of the family of Riordantype 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(j1, m)*A049310(nj, 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/(1x^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+kj, 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



