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%I #15 May 12 2019 02:22:48
%S 1,1,1,1,1,1,2,2,1,1,1,3,2,1,1,3,4,3,2,1,1,1,7,5,3,2,1,1,4,7,8,5,3,2,
%T 1,1,1,14,12,8,5,3,2,1,1
%N Array a(n,m) = Sum_{j=floor((n-1)/2)-m..floor(n-1)/2} binomial(n-j-1,j) read by antidiagonals.
%C A102426 defines an array of binomials in which partial sums of row n yield row a(n,.).
%D Burton, David M., Elementary number theory, McGraw Hill, N.Y., 2002, p. 286.
%e a(n,m) starts in row n=1 as
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
%e 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...
%e 1, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, ...
%e 3, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, ...
%e 1, 7, 12, 13, 13, 13, 13, 13, 13, 13, 13, ...
%e 4, 14, 20, 21, 21, 21, 21, 21, 21, 21, 21, ...
%e 1, 11, 26, 33, 34, 34, 34, 34, 34, 34, 34, ...
%p A175685 := proc(n,m) upl := floor( (n-1)/2) ; add( binomial(n-j-1,j),j=upl-m .. upl) ; end proc: # _R. J. Mathar_, Dec 05 2010
%t a = Table[Table[Sum[Binomial[n -j - 1, j], {j, Floor[(n - 1)/2] - m, Floor[(n - %t 1)/2]}], {n, 0, 10}], {m, 0, 10}];
%t Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}];Flatten[%]
%Y Cf. A000045, A011973, A102426.
%K nonn,tabl,easy
%O 1,7
%A _Roger L. Bagula_, Dec 04 2010
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