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%I A054448 #9 Jul 09 2025 03:54:58
%S A054448 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,
%T A054448 35,8,1,1460,1052,665,349,144,44,9,1,3873,2888,1937,1116,528,196,54,
%U A054448 10,1,10191,7813,5490,3402,1788,768,259,65,11,1,26633,20892,15240,10008
%N A054448 Triangle of partial row sums of triangle A054446(n,m), n >= m >= 0.
%C A054448 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).
%C A054448 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.
%F A054448 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).
%F A054448 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.
%F A054448 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).
%e A054448 {1}; {3,1}; {9,4,1}; {26,14,5,1};...
%e A054448 Fourth row polynomial (n=3): p(3,x)= 26+14*x+5*x^2+x^3
%Y A054448 Cf. A037027, A000045, A000129. Row sums: A054449.
%K A054448 easy,nonn,tabl
%O A054448 0,2
%A A054448 _Wolfdieter Lang_, Apr 27 2000 and May 08 2000