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

1,2

Links

Table of n, a(n) for n=1..33.

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).

Example

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.

Prog

(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}

Crossrefs

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

Sequence in context: A163128 A280924 A295756 * A297840 A102483 A134916

Adjacent sequences: A140044 A140045 A140046 * A140048 A140049 A140050

Keyword

nonn,tabl

Author

Paul D. Hanna, May 12 2008

Status

approved