

A054123


Right Fibonacci rowsum array T(n,k), n >= 0, 0<=k<=n.


8



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
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OFFSET

0,8


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(n1), then x + 1 is in g(n); and
(2) if x is in g(n1) 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^(n1), and T(n,k) is the number of occurrences of 2^(n1k), for k = 0..n1.  Clark Kimberling, Nov 09 2015


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


FORMULA

T(n, 0) = T(n, n) = 1 for n >= 0; T(n, k) = T(n1, k) + T(n2, k1) for k=1, 2, ..., n1, n >= 2.
T(n, k) = T(n1, k1) + U(n1, k) for k=1, 2, ..., floor(n/2), n >= 3, array U as in A011973.


EXAMPLE

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


MATHEMATICA

Clear[t]; t[n_, k_] := t[n, k] = If[k == 0  k == n  k == n1, 1, t[n1, k] + t[n2, k1]]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* JeanFrançois Alcover, Feb 01 2013 *)


PROG

(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


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.
Sequence in context: A108299 A065941 A123320 * A119269 A225630 A129713
Adjacent sequences: A054120 A054121 A054122 * A054124 A054125 A054126


KEYWORD

nonn,tabl,eigen,easy,nice


AUTHOR

Clark Kimberling


EXTENSIONS

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


STATUS

approved



