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A054124 Left Fibonacci row-sum array, n >= 0, 0<=k<=n.

%I A054124 #41 Aug 13 2024 05:05:48
%S A054124 1,1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,2,4,4,1,1,1,2,4,7,5,1,1,1,2,4,8,11,
%T A054124 6,1,1,1,2,4,8,15,16,7,1,1,1,2,4,8,16,26,22,8,1,1,1,2,4,8,16,31,42,29,
%U A054124 9,1,1,1,2,4,8,16,32,57,64,37,10,1,1,1,2
%N A054124 Left Fibonacci row-sum array, n >= 0, 0<=k<=n.
%C A054124 Reflection of array in A054123 about vertical central line.
%C A054124 Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:
%C A054124   (1)  if x is in g(n-1), then x+1 is in g(n); and
%C A054124   (2)  if x is in g(n-1) and x < 2, then x/2 is in g(n).
%C A054124 Then g(1) = {1/1}, g(2) = {1/2,2/1}, g(3) = {1/4,3/2,3/1}, etc. The denominators in g(n) are 2^0, 2^1, ..., 2^(n-1), and T(n,k) is the number of occurrences of 2^k, for k = 0..n-1. - _Clark Kimberling_, Nov 09 2015
%C A054124 Variant of A004070 with an additional column of 1's on the left. - _Jianing Song_, May 30 2022
%H A054124 Reinhard Zumkeller, <a href="/A054124/b054124.txt">Rows n = 0..125 of triangle, flattened</a>
%H A054124 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A054124 T(n, 0) = T(n, n) = 1 for n >= 0; T(n, 1) = 1 for n >= 1; T(n, k) = T(n-1, k-1) + T(n-2, k-1) for k=2, 3, ..., n-1, n >= 3. [Corrected by _Jianing Song_, May 30 2022]
%F A054124 G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = (1-x^2*y) / ((1-x)*(1-x*y-x^2*y)). - _Jianing Song_, May 30 2022
%e A054124 Rows:
%e A054124 1
%e A054124 1 1
%e A054124 1 1 1
%e A054124 1 1 2 1
%e A054124 1 1 2 3 1
%e A054124 ...
%t A054124 t[_, 0|1] = t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-2, k-1]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 25 2013 *)
%o A054124 (Haskell)
%o A054124 a054124 n k = a054124_tabl !! n !! k
%o A054124 a054124_row n = a054124_tabl !! n
%o A054124 a054124_tabl = map reverse a054123_tabl
%o A054124 -- _Reinhard Zumkeller_, May 26 2015
%o A054124 (PARI) A052509(n,k) = sum(m=0, k, binomial(n-k, m));
%o A054124 T(n,k) = if(k==0, 1, A052509(n-1,n-k)) \\ _Jianing Song_, May 30 2022
%Y A054124 Row sums: A000045. Central numbers: 1, 1, 2, 4, 8, ... (A000079).
%Y A054124 First n numbers of n-th column for n >= 1 form the array in A008949.
%K A054124 nonn,tabl,eigen,nice
%O A054124 0,9
%A A054124 _Clark Kimberling_