A110205 Triangle, read by rows, where T(n,k) equals the sum of cubes of numbers < 2^n having exactly k ones in their binary expansion.

1, 9, 27, 73, 368, 343, 585, 3825, 6615, 3375, 4681, 36394, 88536, 86614, 29791, 37449, 332883, 1024002, 1449198, 970677, 250047, 299593, 2979420, 10970133, 20078192, 19714083, 9974580, 2048383, 2396745, 26298405, 112122225, 250021125

Offset

1,2

Comments

Compare to triangle A110200 (sum of squares).

Links

Paul D. Hanna, Rows n = 1..45, flattened.

Formula

T(n, k) = (8^n-1)/7*C(n-3, k-1) + ((2^n-1)*(4^n-1)-(8^n-1)/7)*C(n-3, k-2) + (2^n-1)^3*C(n-3, k-3).

G.f. for row n: ((8^n-1)/7 + ((2^n-1)*(4^n-1)-(8^n-1)/7)*x + (2^n-1)^3*x^2)*(1+x)^(n-3).

Example

Row 4 is formed by sums of cubes of numbers < 2^4:
T(4,1) = 1^3 + 2^3 + 4^3 + 8^3 = 585;
T(4,2) = 3^3 + 5^3 + 6^3 + 9^3 + 10^3 + 12^3 = 3825;
T(4,3) = 7^3 + 11^3 + 13^3 + 14^3 = 6615;
T(4,4) = 15^3 = 3375.
Triangle begins:
1;
9, 27;
73, 368, 343;
585, 3825, 6615, 3375;
4681, 36394, 88536, 86614, 29791;
37449, 332883, 1024002, 1449198, 970677, 250047;
299593, 2979420, 10970133, 20078192, 19714083, 9974580, 2048383;
2396745, 26298405, 112122225, 250021125, 320944275, 239783895, 97221555, 16581375; ...
Row g.f.s are:
row 1: (1 + 2*x + 1*x^2)/(1+x)^2;
row 2: (9 + 36*x + 27*x^2)/(1+x);
row 3: (73 + 368*x + 343*x^2);
row 4: (585 + 3240*x + 3375*x^2)*(1+x).
G.f. for row n is:
((8^n-1)/7 + ((2^n-1)*(4^n-1)-(8^n-1)/7)*x + (2^n-1)^3*x^2)*(1+x)^(n-3).

Prog

(PARI) T(n, k)=(8^n-1)/7*binomial(n-3, k-1)+((2^n-1)*(4^n-1)-(8^n-1)/7)*binomial(n-3, k-2) +(2^n-1)^3*binomial(n-3, k-3)
(PARI) /* Sum of Cubes of numbers<2^n with k 1-bits: */
T(n, k)=local(B=vector(n+1)); if(n<k || k<1, 0, for(m=1, 2^n-1, B[1+sum(i=1, #binary(m), (binary(m))[i])]+=m^3); B[k+1])

Crossrefs

Cf. A110206 (row sums), A110207 (central terms), A023001 (column 1).

Sequence in context: A129957 A328408 A198956 * A319085 A211531 A264959

Adjacent sequences: A110202 A110203 A110204 * A110206 A110207 A110208

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 16 2005

Status

approved