1,2

Compare to triangle A110200 (sum of squares).

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).

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; ...

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).

(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])}

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

Sequence in context: A011923 A029875 A129957 this_sequence A135415 A053702 A036314

Adjacent sequences: A110202 A110203 A110204 this_sequence A110206 A110207 A110208

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 16 2005