A177352 - OEIS

%I #8 Mar 12 2014 16:37:17

%S 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,

%T 20,14,4,34,34,34,34,34,33,26,11,1,55,55,55,55,55,54,46,25,5,89,89,89,

%U 89,89,89,88,79,51,16,1

%N 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.

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

%C 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.

%C 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.

%C Comment _R. J. Mathar_, Dec 20, 2010 (Start):

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

%C 1;  # complement to 2

%C 1,4;  # complement to 4,1

%C 1,5   # complement to 7,3

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

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

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

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

%e 1

%e 1;

%e 2, 2, 1;

%e 3, 3, 2;

%e 5, 5, 5, 4, 1;

%e 8, 8, 8, 7, 3;

%e 13, 13, 13, 13, 12, 7, 1;

%e 21, 21, 21, 21, 20, 14, 4;

%e 34, 34, 34, 34, 34, 33, 26, 11, 1;

%e 55, 55, 55, 55, 55, 54, 46, 25, 5;

%e 89, 89, 89, 89, 89, 89, 88, 79, 51, 16, 1;

%t w[n_, m_, k_] = Binomial[n - (m + k), m + k];

%t t[n_, k_] := Sum[w[n, m, k], {m, 1, Floor[n/2 - k]}];

%t Table[Table[t[n, k], {k, -Floor[n/2 + 1], Floor[n/2 + 1] - 2}], {n, 0,

%t    10}]

%t Flatten[%]

%Y Cf. A177351, A000045

%K nonn,tabf

%O 0,3

%A _Roger L. Bagula_, Dec 10 2010