login
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.
4
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
1,2
COMMENTS
Compare to triangle A110200 (sum of squares).
LINKS
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).
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 *)
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])
(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
(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
CROSSREFS
Cf. A110206 (row sums), A110207 (central terms), A023001 (column 1).
Sequence in context: A373479 A328408 A198956 * A319085 A211531 A264959
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jul 16 2005
STATUS
approved