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A054125 Sum of the arrays in A054123 and A054124.

%I A054125 #22 Aug 15 2025 22:32:01
%S A054125 2,2,2,2,2,2,2,3,3,2,2,4,4,4,2,2,5,6,6,5,2,2,6,9,8,9,6,2,2,7,13,12,12,
%T A054125 13,7,2,2,8,18,19,16,19,18,8,2,2,9,24,30,24,24,30,24,9,2,2,10,31,46,
%U A054125 39,32,39,46,31,10,2,2,11,39,68,65,48,48,65
%N A054125 Sum of the arrays in A054123 and A054124.
%C A054125 Row sums are twice Fibonacci numbers, A006355(n+2).
%F A054125 From _Jianing Song_, May 30 2022: (Start)
%F A054125 T(n,k) = 2 if k = 0 or k = n, A052509(n-1,k) + A052509(n-1,n-k) otherwise.
%F A054125 G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = (1-x^2*y) * (1/((1-x*y)*(1-x-x^2*y)) + 1/((1-x)*(1-x*y-x^2*y))). (End)
%e A054125 Rows:
%e A054125   2;
%e A054125   2,2;
%e A054125   2,2,2;
%e A054125   2,3,3,2;
%e A054125   ...
%o A054125 (PARI) A052509(n,k) = sum(m=0, k, binomial(n-k, m));
%o A054125 T(n,k) = if(k==0 || k==n, 2, A052509(n-1,k) + A052509(n-1,n-k)) \\ _Jianing Song_, May 30 2022
%K A054125 nonn,tabl,eigen
%O A054125 0,1
%A A054125 _Clark Kimberling_