This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A110205 #18 Oct 05 2024 09:40:24
%S A110205 1,9,27,73,368,343,585,3825,6615,3375,4681,36394,88536,86614,29791,
%T A110205 37449,332883,1024002,1449198,970677,250047,299593,2979420,10970133,
%U A110205 20078192,19714083,9974580,2048383,2396745,26298405,112122225,250021125,320944275,239783895,97221555,16581375
%N 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.
%C A110205 Compare to triangle A110200 (sum of squares).
%H A110205 Paul D. Hanna, <a href="/A110205/b110205.txt">Rows n = 1..45, flattened.</a>
%F A110205 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).
%F A110205 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).
%e A110205 Row 4 is formed by sums of cubes of numbers < 2^4:
%e A110205 T(4,1) = 1^3 + 2^3 + 4^3 + 8^3 = 585;
%e A110205 T(4,2) = 3^3 + 5^3 + 6^3 + 9^3 + 10^3 + 12^3 = 3825;
%e A110205 T(4,3) = 7^3 + 11^3 + 13^3 + 14^3 = 6615;
%e A110205 T(4,4) = 15^3 = 3375.
%e A110205 Triangle begins:
%e A110205 1;
%e A110205 9, 27;
%e A110205 73, 368, 343;
%e A110205 585, 3825, 6615, 3375;
%e A110205 4681, 36394, 88536, 86614, 29791;
%e A110205 37449, 332883, 1024002, 1449198, 970677, 250047;
%e A110205 299593, 2979420, 10970133, 20078192, 19714083, 9974580, 2048383;
%e A110205 2396745, 26298405, 112122225, 250021125, 320944275, 239783895, 97221555, 16581375; ...
%e A110205 Row g.f.s are:
%e A110205 row 1: (1 + 2*x + 1*x^2)/(1+x)^2;
%e A110205 row 2: (9 + 36*x + 27*x^2)/(1+x);
%e A110205 row 3: (73 + 368*x + 343*x^2);
%e A110205 row 4: (585 + 3240*x + 3375*x^2)*(1+x).
%e A110205 G.f. for row n is:
%e A110205 ((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).
%t A110205 b[n_, k_]= Binomial[n-3, k];
%t A110205 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]);
%t A110205 A110205[n_, k_]:= If[n<3, T[n,k]/2, T[n,k]];
%t A110205 Table[A110205[n,k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Oct 03 2024 *)
%o A110205 (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)
%o A110205 (PARI) /* Sum of cubes of numbers<2^n with k 1-bits: */
%o A110205 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])
%o A110205 (Magma)
%o A110205 b:= func< n,k | Binomial(n-3, k) >;
%o A110205 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)) >;
%o A110205 [A110205(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 03 2024
%o A110205 (SageMath)
%o A110205 def b(n,k): return binomial(n-3, k)
%o A110205 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))
%o A110205 flatten([[A110205(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Oct 03 2024
%Y A110205 Cf. A110206 (row sums), A110207 (central terms), A023001 (column 1).
%K A110205 nonn,tabl
%O A110205 1,2
%A A110205 _Paul D. Hanna_, Jul 16 2005