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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified January 31 07:41 EST 2023. Contains 359950 sequences. (Running on oeis4.)