A177352 The triangle t(n,k) of the binomial sum as in A177351 in the column index range -floor(n/2)-1 <=k <= floor(n/2)-1.

1, 1, 2, 2, 1, 3, 3, 2, 5, 5, 5, 4, 1, 8, 8, 8, 7, 3, 13, 13, 13, 13, 12, 7, 1, 21, 21, 21, 21, 20, 14, 4, 34, 34, 34, 34, 34, 33, 26, 11, 1, 55, 55, 55, 55, 55, 54, 46, 25, 5, 89, 89, 89, 89, 89, 89, 88, 79, 51, 16, 1

Offset

0,3

Comments

Row sums are 1, 1, 5, 8, 20, 34, 72, 122, 241, 405, 769, 1284, 2375, 3947, 7165,

11866, 21238, 35078, 62094, 102340, 179561,.... which apparently is (n+1)*Fibonacci(n+1)- A129722(n) for even n, and n*Fibonacci(n+1)-A129722(n) for odd n.

The first column is A000045 by construction. The change in the column index range adds the Fibonacci numbers as a first column and removes the trailing zero in the rows compared to A177351.

Comment R. J. Mathar, Dec 20, 2010 (Start):

If we construct the complements of each row's entries with respect to the Fibonacci number of that row, an array

1; # complement to 2

1,4; # complement to 4,1

1,5 # complement to 7,3

1,6,12 # complement to 12,7,1

1,7,17 # complement to 20,14,4

1,8,23,33 # complement to 33,26,11,1

emerges which appears to be related to A038791. (End).

Links

Table of n, a(n) for n=0..60.

Example

1
1;
2, 2, 1;
3, 3, 2;
5, 5, 5, 4, 1;
8, 8, 8, 7, 3;
13, 13, 13, 13, 12, 7, 1;
21, 21, 21, 21, 20, 14, 4;
34, 34, 34, 34, 34, 33, 26, 11, 1;
55, 55, 55, 55, 55, 54, 46, 25, 5;
89, 89, 89, 89, 89, 89, 88, 79, 51, 16, 1;

Mathematica

w[n_, m_, k_] = Binomial[n - (m + k), m + k];
t[n_, k_] := Sum[w[n, m, k], {m, 1, Floor[n/2 - k]}];
Table[Table[t[n, k], {k, -Floor[n/2 + 1], Floor[n/2 + 1] - 2}], {n, 0,
   10}]
Flatten[%]

Crossrefs

Cf. A177351, A000045

Sequence in context: A011373 A321783 A327035 * A210798 A374435 A117501

Adjacent sequences: A177349 A177350 A177351 * A177353 A177354 A177355

Keyword

nonn,tabf

Author

Roger L. Bagula, Dec 10 2010

Status

approved