A121643 a(n) = sum of cubes of the coefficients of x^n in x^(n-3k)*A(x^3)^(n-3k+1), as k varies from 0 to floor(n/3) for n>0, with a(0)=1.

1, 1, 1, 2, 9, 28, 66, 153, 433, 1352, 4104, 12188, 37506, 124155, 422307, 1437135, 5058945, 18255295, 65336580, 232233723, 832793160, 3003712770, 10788778026, 38582183151, 137946936834, 493981065171, 1770660835057, 6347845365236

Offset

0,4

Comments

Defined using a variation of a property of the Catalan sequence: A000108(n) = Sum_{k=0..[n/2]} { [x^n] x^(n-2k)*Catalan(x^2)^(n-2k+1) }^2.

Links

Table of n, a(n) for n=0..27.

Formula

a(n) = Sum_{k=0..[n/3]} { [x^n] x^(n-3k)*A(x^3)^(n-3k+1) }^3 for n>0, with a(0)=1.

Example

a(7) = 153 = 3^3 + 5^3 + 1^3;
a(8) = 433 = 6^3 + 6^3 + 1^3;
a(9) = 1352 = 2^3 + 10^3 + 7^3 + 1^3;
a(10) = 4104 = 6^3 + 15^3 + 8^3 + 1^3.
Initial nonzero coefficients of x^n*A(x^3)^(n+1) are shown below.
Sum of cubes:__1,__1,__1,__2,__9,_28,_66,153,433,1352,4104,..
x^0*A(x^3)^1:__1,__________1,__________1,___________2,____...
x^1*A(x^3)^2:______1,__________2,__________3,____________6,..
x^2*A(x^3)^3:__________1,__________3,__________6,_________...
x^3*A(x^3)^4:______________1,__________4,__________10,____...
x^4*A(x^3)^5:__________________1,__________5,___________15,..
x^5*A(x^3)^6:______________________1,__________6,_________...
x^6*A(x^3)^7:__________________________1,___________7,____...
x^7*A(x^3)^8:______________________________1,____________8,..
x^8*A(x^3)^9:__________________________________1,_________...
x^9*A(x^3)^10:______________________________________1,____...
x^10*A(x^3)^11:__________________________________________1,..

Prog

(PARI) {a(n)=if(n==0, 1, sum(k=0, n\3, polcoeff(x^(n-3*k)*(sum(j=0, k, a(j)*x^(3*j)) +x*O(x^n))^(n-3*k+1), n)^3))}

Crossrefs

Cf. A095892, A000108.

Sequence in context: A303373 A001093 A248658 * A353017 A183376 A131066

Adjacent sequences: A121640 A121641 A121642 * A121644 A121645 A121646

Keyword

nonn

Author

Paul D. Hanna, Aug 13 2006

Status

approved