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%I #7 Feb 08 2022 22:42:20
%S 1,0,1,0,2,1,0,3,0,1,0,4,0,4,1,0,5,0,10,0,1,0,6,0,20,0,6,1,0,7,0,35,0,
%T 21,0,1,0,8,0,56,0,56,0,8,0,1,0,9,0,84,0,126,0,36,0,1,0,10,0,120,0,
%U 252,0,120,0,10,1
%N Triangle by columns, odd columns of Pascal's triangle A007318, otherwise (1, 0, 0, 0, ...).
%C Row sums = a variant of A052950, starting (1, 1, 3, 4, 9, 16, 33, ...); whereas A052950 starts (2, 1, 3, 4, 9, ...).
%C Column 1 of the inverse of A178616 is a signed variant of A065619 prefaced with a 0; where A065619 = (1, 2, 3, 8, 25, 96, 427, ...).
%F Triangle, odd columns of Pascal's triangle; (1, 0, 0, 0, ...) as even columns k.
%F Alternatively, (since A178616 + A162169 - Identity matrix) = Pascal's triangle,
%F we can begin with Pascal's triangle, subtract A162169, then add the Identity
%F matrix to obtain A178616.
%e First few rows of the triangle:
%e 1,
%e 0, 1;
%e 0, 2, 1;
%e 0, 3, 0, 1
%e 0, 4, 0, 4, 1;
%e 0, 5, 0, 10, 0, 1;
%e 0, 6, 0, 20, 0, 6, 1;
%e 0, 7, 0, 35, 0, 21, 0, 1;
%e 0, 8, 0, 56, 0, 56, 0, 8, 1;
%e 0, 9, 0, 84, 0, 126, 0, 36, 0, 1;
%e 0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 1;
%e 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1;
%e ...
%Y Cf. A162109, A065619, A052950, A095704.
%K nonn,tabl
%O 0,5
%A _Gary W. Adamson_, May 30 2010
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