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.

1, 2, 3, 4, 14, 70, 8, 60, 620, 7200, 16, 248, 5208, 123008, 3098760, 32, 1008, 42672, 2032128, 103223568, 5461682688, 64, 4064, 345440, 33032192, 3369214496, 357969864704, 39119789090720, 128, 16320, 2779840, 532684800, 108880217152

Offset: 1

Author

Paul D. Hanna, May 12 2008

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1;
2, 3;
4, 14, 70;
8, 60, 620, 7200;
16, 248, 5208, 123008, 3098760;
32, 1008, 42672, 2032128, 103223568, 5461682688;
64, 4064, 345440, 33032192, 3369214496, 357969864704, 39119789090720; ...
For n=3, since A010060(k) = 0 at k={0,3,5,6}, then
T(3,k) = 0^k + 3^k + 5^k + 6^k for k=0..2;
and since A010060(k) = 1 at k={1,2,4,7}, we also have
T(3,k) = 1^k + 2^k + 4^k + 7^k for k=0..2.
For n=4, since A010060(k) = 0 at k={0,3,5,6,9,10,12,15}, then
T(4,k) = 0^k + 3^k + 5^k + 6^k + 9^k + 10^k + 12^k + 15^k for k=0..3;
and since A010060(k) = 1 at k={1,2,4,7,8,11,13,14}, we also have
T(4,k) = 1^k + 2^k + 4^k + 7^k + 8^k + 11^k + 13^k + 14^k for k=0..3.
		

Crossrefs

Cf. A140048 (main diagonal), A010060, A016290.

Programs

PARI

{T(n,k)=(1/2)*sum(j=0,2^n-1,j^k)}

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}

Formula

T(n,k) = Sum_{j=0..2^n-1, A010060(j)=0 } j^k for k=0..n-1, n>=1; also,

T(n,k) = Sum_{j=0..2^n-1, A010060(j)=1 } j^k for k=0..n-1, n>=1;

where A010060 is the Thue-Morse sequence (identity due to Prouhet).

T(n,0) = 2^n; T(n,1) = 4^n - 2^(n-1); T(n,2) = A016290(n)/2;

T(n,n-1) = A140048(n).