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.
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 13 2006
STATUS
approved