cp's OEIS Frontend

A178616 Triangle by columns, odd columns of Pascal's triangle A007318, otherwise (1, 0, 0, 0, ...).

%I A178616 #7 Feb 08 2022 22:42:20
%S A178616 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 A178616 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 A178616 252,0,120,0,10,1
%N A178616 Triangle by columns, odd columns of Pascal's triangle A007318, otherwise (1, 0, 0, 0, ...).
%C A178616 Row sums = a variant of A052950, starting (1, 1, 3, 4, 9, 16, 33, ...); whereas A052950 starts (2, 1, 3, 4, 9, ...).
%C A178616 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 A178616 Triangle, odd columns of Pascal's triangle; (1, 0, 0, 0, ...) as even columns k.
%F A178616 Alternatively, (since A178616 + A162169 - Identity matrix) = Pascal's triangle,
%F A178616 we can begin with Pascal's triangle, subtract A162169, then add the Identity
%F A178616 matrix to obtain A178616.
%e A178616 First few rows of the triangle:
%e A178616   1,
%e A178616   0,  1;
%e A178616   0,  2, 1;
%e A178616   0,  3, 0,   1
%e A178616   0,  4, 0,   4, 1;
%e A178616   0,  5, 0,  10, 0,   1;
%e A178616   0,  6, 0,  20, 0,   6, 1;
%e A178616   0,  7, 0,  35, 0,  21, 0,   1;
%e A178616   0,  8, 0,  56, 0,  56, 0,   8, 1;
%e A178616   0,  9, 0,  84, 0, 126, 0,  36, 0,  1;
%e A178616   0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 1;
%e A178616   0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1;
%e A178616   ...
%Y A178616 Cf. A162109, A065619, A052950, A095704.
%K A178616 nonn,tabl
%O A178616 0,5
%A A178616 _Gary W. Adamson_, May 30 2010