%I #4 Mar 30 2012 18:36:58
%S 1,1,1,2,9,28,66,153,433,1352,4104,12188,37506,124155,422307,1437135,
%T 5058945,18255295,65336580,232233723,832793160,3003712770,10788778026,
%U 38582183151,137946936834,493981065171,1770660835057,6347845365236
%N 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.
%C 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.
%F 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.
%e a(7) = 153 = 3^3 + 5^3 + 1^3;
%e a(8) = 433 = 6^3 + 6^3 + 1^3;
%e a(9) = 1352 = 2^3 + 10^3 + 7^3 + 1^3;
%e a(10) = 4104 = 6^3 + 15^3 + 8^3 + 1^3.
%e Initial nonzero coefficients of x^n*A(x^3)^(n+1) are shown below.
%e Sum of cubes:__1,__1,__1,__2,__9,_28,_66,153,433,1352,4104,..
%e x^0*A(x^3)^1:__1,__________1,__________1,___________2,____...
%e x^1*A(x^3)^2:______1,__________2,__________3,____________6,..
%e x^2*A(x^3)^3:__________1,__________3,__________6,_________...
%e x^3*A(x^3)^4:______________1,__________4,__________10,____...
%e x^4*A(x^3)^5:__________________1,__________5,___________15,..
%e x^5*A(x^3)^6:______________________1,__________6,_________...
%e x^6*A(x^3)^7:__________________________1,___________7,____...
%e x^7*A(x^3)^8:______________________________1,____________8,..
%e x^8*A(x^3)^9:__________________________________1,_________...
%e x^9*A(x^3)^10:______________________________________1,____...
%e x^10*A(x^3)^11:__________________________________________1,..
%o (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))}
%Y Cf. A095892, A000108.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Aug 13 2006