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A054123 Right Fibonacci row-sum array T(n,k), n >= 0, 0<=k<=n.

%I A054123 #42 Aug 13 2024 04:58:58
%S A054123 1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,4,2,1,1,1,5,7,4,2,1,1,1,6,11,8,4,2,
%T A054123 1,1,1,7,16,15,8,4,2,1,1,1,8,22,26,16,8,4,2,1,1,1,9,29,42,31,16,8,4,2,
%U A054123 1,1,1,10,37,64,57,32,16,8,4,2,1,1,1,11,46,93,99,63,32,16,8,4,2,1,1,1,12,56
%N A054123 Right Fibonacci row-sum array T(n,k), n >= 0, 0<=k<=n.
%C A054123 Variant of A052509 with an additional diagonal of 1's. - _R. J. Mathar_, Oct 12 2011
%C A054123 Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:
%C A054123   (1)  if x is in g(n-1), then x + 1 is in g(n); and
%C A054123   (2)  if x is in g(n-1) and x < 2, then x/2 is in g(n).
%C A054123 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^(n-1-k), for k = 0..n-1. - _Clark Kimberling_, Nov 09 2015
%C A054123 G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = (1-x^2*y) / ((1-x*y)*(1-x-x^2*y)). - _Jianing Song_, May 30 2022
%H A054123 Reinhard Zumkeller, <a href="/A054123/b054123.txt">Rows n = 0..125 of triangle, flattened</a>
%H A054123 S. Sivasubramanian, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i2p10">Signed excedance enumeration in the hyperoctahedral group</a> El. J. Combinat. 21 (1) (2014) #P2.10, Remark 16.
%H A054123 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A054123 T(n, 0) = T(n, n) = 1 for n >= 0; T(n, n-1) = 1 for n >= 1; T(n, k) = T(n-1, k) + T(n-2, k-1) for k=1, 2, ..., n-2, n >= 3. [Corrected by _Jianing Song_, May 30 2022]
%F A054123 T(n, k) = T(n-1, k-1) + U(n-1, k) for k=1, 2, ..., floor(n/2), n >= 3, array U as in A011973.
%e A054123 Rows:
%e A054123 1
%e A054123 1 1
%e A054123 1 1 1
%e A054123 1 2 1 1
%e A054123 1 3 2 1 1
%e A054123 1 4 4 2 1 1
%e A054123 1 5 7 4 2 1 1
%t A054123 Clear[t]; t[n_, k_] := t[n, k] = If[k == 0 || k == n || k == n-1, 1, t[n-1, k] + t[n-2, k-1]]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 01 2013 *)
%o A054123 (Haskell)
%o A054123 a054123 n k = a054123_tabl !! n !! k
%o A054123 a054123_row n = a054123_tabl !! n
%o A054123 a054123_tabl = [1] : [1, 1] : f [1] [1, 1] where
%o A054123    f us vs = ws : f vs ws where
%o A054123              ws = zipWith (+) (0 : init us ++ [0, 0]) (vs ++ [1])
%o A054123 -- _Reinhard Zumkeller_, May 26 2015
%o A054123 (PARI) A052509(n,k) = sum(m=0, k, binomial(n-k, m));
%o A054123 T(n,k) = if(k==n, 1, A052509(n-1,k)) \\ _Jianing Song_, May 30 2022
%Y A054123 Reflection of array in A054124 about vertical central line.
%Y A054123 Row sums: 1, 2, 3, 5, 8, 13, ... (Fibonacci numbers, A000045). Central numbers: 1, 1, 2, 4, 8, ... (binary powers, A000079). Cf. A011973.
%Y A054123 Cf. A129713.
%K A054123 nonn,tabl,eigen,easy,nice
%O A054123 0,8
%A A054123 _Clark Kimberling_
%E A054123 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 05 2003