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

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, 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, 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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl
• eigen
• easy
• nice

Comments

Variant of A052509 with an additional diagonal of 1's. - R. J. Mathar, Oct 12 2011

Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:

(1) if x is in g(n-1), then x + 1 is in g(n); and

(2) if x is in g(n-1) and x < 2, then x/2 is in g(n).

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

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

Examples

			Rows:
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
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• S. Sivasubramanian, Signed excedance enumeration in the hyperoctahedral group El. J. Combinat. 21 (1) (2014) #P2.10, Remark 16.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Reflection of array in A054124 about vertical central line.

Row sums: 1, 2, 3, 5, 8, 13, ... (Fibonacci numbers, A000045). Central numbers: 1, 1, 2, 4, 8, ... (binary powers, A000079). Cf. A011973.

Cf. A129713.

Programs

Haskell

a054123 n k = a054123_tabl !! n !! k
a054123_row n = a054123_tabl !! n
a054123_tabl = [1] : [1, 1] : f [1] [1, 1] where
   f us vs = ws : f vs ws where
             ws = zipWith (+) (0 : init us ++ [0, 0]) (vs ++ [1])
-- Reinhard Zumkeller, May 26 2015

Mathematica

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 *)

PARI

A052509(n,k) = sum(m=0, k, binomial(n-k, m));
T(n,k) = if(k==n, 1, A052509(n-1,k)) \\ Jianing Song, May 30 2022

Formula

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]

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.

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 05 2003