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A177351 Triangle t(n,k)= sum_{m=1..floor(n/2-k)} binomial(n-m-k,m+k), -floor(n/2) <= k <= floor(n/2), read by rows.

%I A177351 #8 Mar 12 2014 16:37:17
%S A177351 0,0,2,1,0,3,2,0,5,5,4,1,0,8,8,7,3,0,13,13,13,12,7,1,0,21,21,21,20,14,
%T A177351 4,0,34,34,34,34,33,26,11,1,0,55,55,55,55,54,46,25,5,0,89,89,89,89,89,
%U A177351 88,79,51,16,1,0
%N A177351 Triangle t(n,k)= sum_{m=1..floor(n/2-k)} binomial(n-m-k,m+k), -floor(n/2) <= k <= floor(n/2), read by rows.
%C A177351 Row sums are 0, 0, 3, 5, 15, 26, 59, 101, 207, 350, 680,...
%C A177351 The first column is essentially A000045, and the other columns also join the Fibonacci sequence after some transient initial terms.
%e A177351 0;
%e A177351 0;
%e A177351 2, 1, 0;
%e A177351 3, 2, 0;
%e A177351 5, 5, 4, 1, 0;
%e A177351 8, 8, 7, 3, 0;
%e A177351 13, 13, 13, 12, 7, 1, 0;
%e A177351 21, 21, 21, 20, 14, 4, 0;
%e A177351 34, 34, 34, 34, 33, 26, 11, 1, 0;
%e A177351 55, 55, 55, 55, 54, 46, 25, 5, 0;
%e A177351 89, 89, 89, 89, 89, 88, 79, 51, 16, 1, 0;
%t A177351 w[n_, m_, k_] = Binomial[n - (m + k), m + k];
%t A177351 t[n_, k_] := Sum[w[n, m, k], {m, 1, Floor[n/2 - k]}];
%t A177351 Table[Table[t[n, k], {k, -Floor[n/2], Floor[n/2]}], {n, 0, 10}];
%t A177351 Flatten[%]
%Y A177351 Cf. A000045
%K A177351 nonn,tabf
%O A177351 0,3
%A A177351 _Roger L. Bagula_, Dec 10 2010