A110206
Row sums of triangle A110205, where A110205(n,k) equals the sum of cubes of numbers < 2^n having exactly k ones in their binary expansion.
Original entry on oeis.org
1, 36, 784, 14400, 246016, 4064256, 66064384, 1065369600, 17112825856, 274341298176, 4393752592384, 70334388633600, 1125625045712896, 18012199553335296, 288212784234102784, 4611545282012774400
Offset: 1
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[Binomial(2^n,2)^2: n in [1..30]]; // G. C. Greubel, Oct 02 2024
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Binomial[2^Range[30], 2]^2 (* G. C. Greubel, Oct 02 2024 *)
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a(n)=polcoeff(x*(1+8*x)/((1-4*x)*(1-8*x)*(1-16*x)+x*O(x^n)),n)
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def A110206(n): return binomial(2^n, 2)^2
[A110206(n) for n in range(1,31)] # G. C. Greubel, Oct 02 2024
A110200
Triangle, read by rows, where T(n,k) equals the sum of squares of numbers < 2^n having exactly k ones in their binary expansion.
Original entry on oeis.org
1, 5, 9, 21, 70, 49, 85, 395, 535, 225, 341, 1984, 3906, 3224, 961, 1365, 9429, 24066, 29274, 17241, 3969, 5461, 43434, 135255, 215900, 188595, 86106, 16129, 21845, 196095, 717825, 1412275, 1628175, 1106445, 411995, 65025, 87381, 872788
Offset: 1
Row 4 is formed by sums of squares of numbers < 2^4:
T(4,1) = 1^2 + 2^2 + 4^2 + 8^2 = 85;
T(4,2) = 3^2 + 5^2 + 6^2 + 9^2 + 10^2 + 12^2 = 395;
T(4,3) = 7^2 + 11^2 + 13^2 + 14^2 = 535;
T(4,4) = 15^2 = 225.
Triangle begins:
1;
5, 9;
21, 70, 49;
85, 395, 535, 225;
341, 1984, 3906, 3224, 961;
1365, 9429, 24066, 29274, 17241, 3969;
5461, 43434, 135255, 215900, 188595, 86106, 16129;
21845, 196095, 717825, 1412275, 1628175, 1106445, 411995, 65025;
87381, 872788, 3662848, 8541876, 12197570, 10974236, 6095208, 1915228, 261121; ...
Row g.f.s are:
row 1: (1 + 1*x)/(1+x);
row 2: (5 + 9*x);
row 3: (21 + 49*x)*(1+x);
row 4: (85 + 225*x)*(1+x)^2.
G.f. for row n is:
((4^n-1)/3 + (2^n-1)^2*x)*(1+x)^(n-2).
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T(n,k)=(4^n-1)/3*binomial(n-2,k-1)+(2^n-1)^2*binomial(n-2,k-2)
for(n=1, 15, for(k=1, n, print1(T(n, k), ", ")); print(""))
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/* Using G.f. of A(x,y): */
T(n,k)=my(X=x+x*O(x^n),Y=y+y*O(y^k));if(n
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/* Sum of Squares of numbers<2^n with k 1-bits: */
T(n,k)=my(B=vector(n+1));if(n
A110207
a(n) = sum of cubes of numbers < 2^n having exactly floor(n/2) + 1 ones in their binary expansion.
Original entry on oeis.org
1, 27, 368, 6615, 88536, 1449198, 20078192, 320944275, 4584724120, 72867715074, 1064153845776, 16896536592390, 250629464211504, 3980364331323996, 59709362473177824, 948742244639103915, 14352114907032903000
Offset: 1
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b:= func< n,k | Binomial(n-3, Floor(n/2) - k) >;
A110207:= func< n | (8^n-1)/7*(b(n,0) - b(n,1)) + (2^n-1)^2*((2^n+1)*b(n,1) + (2^n-1)*b(n,2)) >;
[A110207(n): n in [1..30]]; // G. C. Greubel, Oct 03 2024
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b[n_, k_]:= Binomial[n-3, Floor[n/2]-k];
f[n_]:= (8^n-1)/7*(b[n,0] - b[n,1]) + (2^n-1)^2*((2^n+1)*b[n,1] + (2^n - 1)*b[n,2]);
A110207[n_]:= If[n<3, f[n]/2, f[n]];
Table[A110207[n], {n,30}] (* G. C. Greubel, Oct 03 2024 *)
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{a(n)=(8^n-1)/7*binomial(n-3,n\2)+((2^n-1)*(4^n-1)-(8^n-1)/7)*binomial(n-3,n\2-1) +(2^n-1)^3*binomial(n-3,n\2-2)}
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def b(n,k): return binomial(n-3, (n//2) - k)
def A110207(n): return (8^n-1)/7*(b(n,0) - b(n,1)) + (2^n-1)^2*((2^n+1)*b(n,1) + (2^n-1)*b(n,2))
[A110207(n) for n in range(1,31)] # G. C. Greubel, Oct 03 2024
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