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%I #10 Jan 26 2014 00:32:31
%S 1,2,1,5,3,1,12,9,4,1,29,24,14,5,1,70,62,42,20,6,1,169,156,118,67,27,
%T 7,1,408,387,316,205,100,35,8,1,985,951,821,588,332,142,44,9,1,2378,
%U 2323,2088,1614,1020,509,194,54,10,1,5741,5652,5232,4290,2966,1671,747,257
%N Triangle of partial row sums of triangle A037027(n,m), n >= m >= 0 (Fibonacci convolution triangle).
%C 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)/(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).
%F a(n, m)=sum(A037027(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n<m, (sequence of partial row sums in columns m).
%F Column m recursion: a(n, m)= sum(a(j-1, m)*A037027(n-j, 0), j=m..n) + A037027(n, m), n >= m >= 0, a(n, m) := 0 if n<m.
%F G.f. for column m: Pell(x)*(x*Fib(x))^m, m >= 0, with Fib(x) = g.f. A000045(n+1) and Pell(x) = g.f. A000129(n+1).
%F T(n,0) = 2*T(n-1,0) + T(n-2,0), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k) for k>0, T(0,0) = 1, T(1,0) = 2, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Jan 26 2014
%e {1}; {2,1}; {5,3,1}; {12,9,4,1};...
%e Fourth row polynomial (n=3): p(3,x)= 12+9*x+4*x^2+x^3
%Y Cf. A037027, A000045, A000129. Row sums: A054447(n).
%K easy,nonn,tabl
%O 0,2
%A _Wolfdieter Lang_, Apr 27 2000 and May 08 2000
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