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%I #8 Jan 26 2019 11:09:10
%S 1,0,1,2,8,34,162,822,4365,23956,134814,773746,4511693,26652346,
%T 159170385,959412290,5829083420,35661048886,219491344362,
%U 1358204062536,8444658457530,52729475008690,330518562341537,2078987880103170
%N Diagonal sums of number triangle A109970.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%F a(0) = 1, a(n) = Sum_{k=0..n-1} (k/(n-k))*binomial(3n-4k-1, n-2k), n>0. [corrected by _Michel Marcus_, Jan 25 2019]
%o (PARI) a(n) = if (n==0, 1, sum(k=0, n-1, (k/(n-k))*binomial(3*n-4*k-1, n-2*k))); \\ _Michel Marcus_, Jan 25 2019
%K easy,nonn
%O 0,4
%A _Paul Barry_, Jul 06 2005