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

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, 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, 9, 1, 1, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl
• eigen
• nice

Comments

Reflection of array in A054123 about vertical central line.

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^k, for k = 0..n-1. - Clark Kimberling, Nov 09 2015

Variant of A004070 with an additional column of 1's on the left. - Jianing Song, May 30 2022

Examples

			Rows:
1
1 1
1 1 1
1 1 2 1
1 1 2 3 1
...
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Row sums: A000045. Central numbers: 1, 1, 2, 4, 8, ... (A000079).

First n numbers of n-th column for n >= 1 form the array in A008949.

Programs

Haskell

a054124 n k = a054124_tabl !! n !! k
a054124_row n = a054124_tabl !! n
a054124_tabl = map reverse a054123_tabl
-- Reinhard Zumkeller, May 26 2015

Mathematica

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 *)

PARI

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

Formula

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]

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