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, 320944275, 239783895, 97221555, 16581375

Offset: 1

Author

Paul D. Hanna, Jul 16 2005

Keywords

• nonn
• tabl

Comments

Compare to triangle A110200 (sum of squares).

Examples

			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).
		

Links

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

Crossrefs

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

Programs

Magma

b:= func< n,k | Binomial(n-3, k) >;
A110205:= func< n,k | (8^n-1)/7*(b(n,k-1) -b(n,k-2)) +(2^n-1)^2*((2^n+1)*b(n,k-2) +(2^n-1)*b(n,k-3)) >;
[A110205(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2024

Mathematica

b[n_, k_]= Binomial[n-3, k];
T[n_, k_]:= (8^n-1)/7*(b[n,k-1] -b[n,k-2]) + (2^n-1)^2*((2^n+1)*b[n,k-2] + (2^n-1)*b[n,k-3]);
A110205[n_, k_]:= If[n<3, T[n,k]/2, T[n,k]];
Table[A110205[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 03 2024 *)

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
				

SageMath

def b(n,k): return binomial(n-3, k)
def A110205(n,k): return (8^n-1)/7*(b(n,k-1) - b(n,k-2)) + (2^n-1)^2*((2^n+1)*b(n,k-2) + (2^n-1)*b(n,k-3))
flatten([[A110205(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 03 2024

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).