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 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified January 18 13:30 EST 2020. Contains 331007 sequences. (Running on oeis4.)