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%I #5 Sep 20 2019 04:50:54
%S 2,2,16,4,32,72,10,80,216,256,28,224,648,1024,800,84,672,2016,3584,
%T 4000,2304,264,2112,6480,12288,16000,13824,6272,858,6864,21384,42240,
%U 60000,62208,43904,16384,2860,22880,72072,146432,220000,253440,219520,131072
%N Triangle read by rows: T(n,k)=k^3*2^k*binomial(2n-k,n-k)/(2n-k) (1<=k<=n).
%C T(n,1) = 2*Catalan(n-1) = 2*A000108(n-1); T(n,n) = 2^n*n^2 = A007758(n).
%C Row sums yield A112328.
%H F. Ruskey, <a href="https://doi.org/10.1137/0601007">Average shape of binary trees</a>, SIAM J. Alg. Disc. Meth., 1, 1980, 43-50 (Eq. (8)).
%e Triangle starts:
%e 2;
%e 2,16;
%e 4,32,72;
%e 10,80,216,256;
%p T:=proc(n,k) if k<2*n then k^3*2^k*binomial(2*n-k,n-k)/(2*n-k) else 0 fi end: for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%Y Cf. A000108, A007758, A112328.
%K nonn,tabl
%O 1,1
%A _Emeric Deutsch_, Sep 04 2005