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A140047 Triangle, read by rows: T(n,k) = (1/2)*Sum_{j=0..2^n-1} j^k for k=0..n-1, n>=1; related to the Prouhet-Tarry-Escott problem.

%I A140047 #2 Mar 30 2012 18:37:10
%S A140047 1,2,3,4,14,70,8,60,620,7200,16,248,5208,123008,3098760,32,1008,42672,
%T A140047 2032128,103223568,5461682688,64,4064,345440,33032192,3369214496,
%U A140047 357969864704,39119789090720,128,16320,2779840,532684800,108880217152
%N A140047 Triangle, read by rows: T(n,k) = (1/2)*Sum_{j=0..2^n-1} j^k for k=0..n-1, n>=1; related to the Prouhet-Tarry-Escott problem.
%F A140047 T(n,k) = Sum_{j=0..2^n-1, A010060(j)=0 } j^k for k=0..n-1, n>=1; also,
%F A140047 T(n,k) = Sum_{j=0..2^n-1, A010060(j)=1 } j^k for k=0..n-1, n>=1;
%F A140047 where A010060 is the Thue-Morse sequence (identity due to Prouhet).
%F A140047 T(n,0) = 2^n; T(n,1) = 4^n - 2^(n-1); T(n,2) = A016290(n)/2;
%F A140047 T(n,n-1) = A140048(n).
%e A140047 Triangle begins:
%e A140047 1;
%e A140047 2, 3;
%e A140047 4, 14, 70;
%e A140047 8, 60, 620, 7200;
%e A140047 16, 248, 5208, 123008, 3098760;
%e A140047 32, 1008, 42672, 2032128, 103223568, 5461682688;
%e A140047 64, 4064, 345440, 33032192, 3369214496, 357969864704, 39119789090720; ...
%e A140047 For n=3, since A010060(k) = 0 at k={0,3,5,6}, then
%e A140047 T(3,k) = 0^k + 3^k + 5^k + 6^k for k=0..2;
%e A140047 and since A010060(k) = 1 at k={1,2,4,7}, we also have
%e A140047 T(3,k) = 1^k + 2^k + 4^k + 7^k for k=0..2.
%e A140047 For n=4, since A010060(k) = 0 at k={0,3,5,6,9,10,12,15}, then
%e A140047 T(4,k) = 0^k + 3^k + 5^k + 6^k + 9^k + 10^k + 12^k + 15^k for k=0..3;
%e A140047 and since A010060(k) = 1 at k={1,2,4,7,8,11,13,14}, we also have
%e A140047 T(4,k) = 1^k + 2^k + 4^k + 7^k + 8^k + 11^k + 13^k + 14^k for k=0..3.
%o A140047 (PARI) {T(n,k)=(1/2)*sum(j=0,2^n-1,j^k)}
%o A140047 (PARI) {T(n,k)=local(Tnk=0);for(j=0,2^n-1,if(subst(Pol(binary(j)),x,1)%2==0,Tnk+=j^k));Tnk}
%Y A140047 Cf. A140048 (main diagonal), A010060, A016290.
%K A140047 nonn,tabl
%O A140047 1,2
%A A140047 _Paul D. Hanna_, May 12 2008