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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.

%I A177352 #10 Jun 29 2025 21:41:28
%S A177352 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 A177352 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 A177352 89,89,89,88,79,51,16,1
%N 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.
%C A177352 Row sums are 1, 1, 5, 8, 20, 34, 72, 122, 241, 405, 769, 1284, 2375, 3947, 7165,
%C A177352 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 A177352 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 A177352 Comment _R. J. Mathar_, Dec 20 2010 (Start):
%C A177352 If we construct the complements of each row's entries with respect to the Fibonacci number of that row, an array
%C A177352 1;  # complement to 2
%C A177352 1,4;  # complement to 4,1
%C A177352 1,5   # complement to 7,3
%C A177352 1,6,12 # complement to 12,7,1
%C A177352 1,7,17 # complement to 20,14,4
%C A177352 1,8,23,33 # complement to 33,26,11,1
%C A177352 emerges which appears to be related to A038791. (End).
%e A177352 1
%e A177352 1;
%e A177352 2, 2, 1;
%e A177352 3, 3, 2;
%e A177352 5, 5, 5, 4, 1;
%e A177352 8, 8, 8, 7, 3;
%e A177352 13, 13, 13, 13, 12, 7, 1;
%e A177352 21, 21, 21, 21, 20, 14, 4;
%e A177352 34, 34, 34, 34, 34, 33, 26, 11, 1;
%e A177352 55, 55, 55, 55, 55, 54, 46, 25, 5;
%e A177352 89, 89, 89, 89, 89, 89, 88, 79, 51, 16, 1;
%t A177352 w[n_, m_, k_] = Binomial[n - (m + k), m + k];
%t A177352 t[n_, k_] := Sum[w[n, m, k], {m, 1, Floor[n/2 - k]}];
%t A177352 Table[Table[t[n, k], {k, -Floor[n/2 + 1], Floor[n/2 + 1] - 2}], {n, 0,
%t A177352    10}]
%t A177352 Flatten[%]
%Y A177352 Cf. A177351, A000045
%K A177352 nonn,tabf
%O A177352 0,3
%A A177352 _Roger L. Bagula_, Dec 10 2010