A171804 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).

1, 1, 5, 33, 252, 2090, 18299, 166450, 1557595, 14898228, 145003996, 1431487820, 14299208690, 144262270360, 1467857359738, 15045486643137, 155208575698230, 1610201799670560, 16788969497000365, 175838914655128068

Offset: 0

Author

Paul D. Hanna, Dec 20 2009

Keywords

• nonn
• easy

Examples

			From _Peter Bala_, Nov 12 2024: (Start)
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + ...
I(P(x)) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 + 2816*x^7 + ...
I^2(P(x)) = 1 + x + 4*x^2 + 20*x^3 + 115*x^4 + 714*x^5 + 4669*x^6 + 31671*x^7 + ...
I^3(P(x)) = 1 + x + 5*x^2 + 33*x^3 + 252*x^4 + 2090*x^5 + 18299*x^6 + 166450*x^7 + ... = the g.f. A(x). (End)
		

Crossrefs

Cf. A000041, A010815, A109085, A145268, A171802, A171803, A171805.

Programs

PARI

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

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)}

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)}

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)}

Formula

G.f. satisfies

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

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

(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.

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

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

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

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

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

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

(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)) + ...

(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 + ...

From Peter Bala, Nov 12 2024: (Start)

A(x) = ( 1/x * series_reversion(x/P(x)^3) )^(1/3).

A(x) = the third iterate I^3(P(x)), where the operator I is defined by I(f(x)) = 1/x * series_reversion(x/f(x)). See the Example section. (Note that I(P(x)) is the g.f. of A109085 and I^2(P(x)) is the g.f. of A171802.) (End)