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%I #23 Jul 29 2013 12:50:08
%S 1,1,1,2,1,1,3,1,1,3,4,1,4,6,5,1,1,10,10,6,1,1,5,20,15,7,1,6,15,35,21,
%T 8,1,1,21,35,56,28,9,1,1,7,56,70,84,36,10,1,8,28,126,126,120,45,11,1,
%U 1,36,84,252,210,165,55,12,1,1,9,120,210,462,330,220,66,13,1
%N Triangle of falling diagonals of A011973 (with rows displayed as centered text).
%C Row sums are A005314 with offset = 1.
%F r(n) = binomial(n-floor((4n+15-6k+(-1)^k)/12), n-floor((4n+15-6k+(-1)^k)/12)-floor((2n-1)/3)+k-1), {k,1,floor((2n+2)/3)}.
%F R(n) = binomial(n-Floor((k+1)/2), n-Floor((3k-1)/2)), {k,1,floor((2n+2)/3)} - gives the terms of each row in reverse order.
%e First 11 triangle rows are :
%e 1
%e 1, 1
%e 2, 1
%e 1, 3, 1
%e 1, 3, 4, 1
%e 4, 6, 5, 1
%e 1, 10, 10, 6, 1
%e 1, 5, 20, 15, 7, 1
%e 6, 15, 35, 21, 8, 1
%e 1, 21, 35, 56, 28, 9, 1
%e 1, 7, 56, 70, 84, 36, 10, 1
%t Table[Reverse[Table[Binomial[n - Floor[(k + 1)/2], n - Floor[(3 k - 1)/2]], {k, Floor[(2 n + 2)/3]}]], {n, 13}] (* _T. D. Noe_, Jul 25 2013 *)
%t Column[Table[Binomial[n - Floor[(4 n + 15 - 6 k + (-1)^k)/12], n - Floor[(4 n + 15 - 6 k + (-1)^k)/12] - Floor[(2 n - 1)/3] + k - 1], {n, 1, 25}, {k, 1, Floor[(2 n + 2)/3]}]] (* _John Molokach_, Jul 25 2013 *)
%Y Cf. A005314, A227300, A001973, A000045, A004396.
%K nonn,tabf
%O 1,4
%A _John Molokach_, Jul 21 2013
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