0,4

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.

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.

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,..

(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))}

Cf. A095892, A000108.

Sequence in context: A155472 A100293 A001093 this_sequence A131066 A058877 A026087

Adjacent sequences: A121640 A121641 A121642 this_sequence A121644 A121645 A121646

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 13 2006