login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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<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])

CROSSREFS

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

Sequence in context: A129957 A328408 A198956 * A319085 A135415 A211531

Adjacent sequences:  A110202 A110203 A110204 * A110206 A110207 A110208

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jul 16 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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