A171804 - OEIS

%I #11 Nov 24 2012 16:43:38

%S 1,1,5,33,252,2090,18299,166450,1557595,14898228,145003996,1431487820,

%T 14299208690,144262270360,1467857359738,15045486643137,

%U 155208575698230,1610201799670560,16788969497000365,175838914655128068

%N G.f. satisfies: A(x) = P(x*A(x)^3) where A(x/P(x)^3) = P(x) is the g.f. for Partition numbers (A000041).

%F G.f. satisfies

%F (1) A(x) = 1/Product_{k>0} (1-x^k*A(x)^3).

%F (2) A(x) = Sum_{n>=0} x^n*A(x)^(3*n) / Product_{k=1..n} (1-x^k*A(x)^(3*k)).

%F (3) A(x) = Sum_{n>=0} x^(n^2)*A(x)^(3*n^2) / Product_{k=1..n} (1-x^k*A(x)^(3*k))^2.

%e G.f.: A(x) = 1 + x + 5*x^2 + 33*x^3 + 252*x^4 + 2090*x^5 +...

%e G.f. satisfies A(x/P(x)^3) = P(x) where:

%e P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...

%e and x/P(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 +...

%e Also, g.f. A = A(x) satisfies:

%e (1) A(x) = 1/((1-x*A^3) * (1-x^2*A^6) * (1-x^3*A^9) * (1-x^4*A^12)...).

%e (2) A(x) = 1 + x*A^3/(1-x*A^3) + x^2*A^6/((1-x*A^3)*(1-x^2*A^6)) + x^3*A^9/((1-x*A^3)*(1-x^2*A^6)*(1-x^3*A^9)) +...

%e (3) A(x) = 1 + x*A^3/(1-x*A^3)^2 + x^4*A^12/((1-x*A^3)*(1-x^2*A^6))^2 + x^9*A^27/((1-x*A^3)*(1-x^2*A^6)*(1-x^3*A^9))^2 +...

%o (PARI) a(n)=polcoeff((1/x*serreverse(x*eta(x+x*O(x^n))^3))^(1/3), n)

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*(A^3+x*O(x^n))^k))); polcoeff(A, n)}

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^(3*m)/prod(k=1, m, (1-x^k+x*O(x^n))))); polcoeff(A, n)}

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(3*m)/prod(k=1, m, (1-x^k)*(1-x^k*A^3+x*O(x^n))))); polcoeff(A, n)}

%Y Cf. A171802, A171803, A171805, A109085, A145268, A000041.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 20 2009