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A229032 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).

%I A229032 #14 Sep 30 2015 10:37:41
%S A229032 1,2,0,3,4,0,4,16,0,0,5,40,16,0,0,6,80,96,0,0,0,7,140,336,64,0,0,0,8,
%T A229032 224,896,512,0,0,0,0,9,336,2016,2304,256,0,0,0,0,10,480,4032,7680,
%U A229032 2560,0,0,0,0,0,11,660,7392,21120,14080,1024,0,0,0,0,0
%N A229032 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).
%C A229032 Row n is the sum of the convolution of A089627(p,i) with A089627(n-p,i), for p=0,1,...,n.
%H A229032 Rui Duarte and António Guedes de Oliveira, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Duarte/duarte7.html">A Famous Identity of Hajós in Terms of Sets</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.1.
%F A229032 T(n,k) = 4^k * C(n+1, 2*k+1).
%F A229032 T(n,k) = sum(p=0..n, sum(i=0..k, C(p,i)*C(p-i, i)*C(n-p,k-i)*C(n-p-k+i, k-i))).
%F A229032 T(n,k) = A085841(n/2,k), if n is even.
%F A229032 T(n,k) = 2^k * A105070(n,k).
%e A229032 Triangle:
%e A229032 1
%e A229032 2, 0
%e A229032 3, 4, 0
%e A229032 4, 16, 0, 0
%e A229032 5, 40, 16, 0, 0
%e A229032 6, 80, 96, 0, 0, 0
%e A229032 7, 140, 336, 64, 0, 0, 0
%e A229032 8, 224, 896, 512, 0, 0, 0, 0
%e A229032 9, 336, 2016, 2304, 256, 0, 0, 0, 0
%e A229032 10, 480, 4032, 7680, 2560, 0, 0, 0, 0, 0
%e A229032 11, 660, 7392, 21120, 14080, 1024, 0, 0, 0, 0, 0
%K A229032 nonn,tabl
%O A229032 0,2
%A A229032 _Rui Duarte_ and António Guedes de Oliveira, Sep 11 2013