A213639 G.f. A(x) satisfies x = A( x - A(x)^3/x ).

1, 1, 5, 38, 357, 3832, 45189, 572378, 7676653, 107971691, 1581714400, 24012849880, 376361077578, 6071985730614, 100602798234000, 1708558136679750, 29698002444820760, 527661478169200755, 9573199146196780335, 177192815265794698364, 3343432166097650920872

Offset: 1

Author

Paul D. Hanna, Jun 16 2012

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 38*x^4 + 357*x^5 + 3832*x^6 + 45189*x^7 +...
Related series:
A(x)^3/x = x^2 + 3*x^3 + 18*x^4 + 145*x^5 + 1389*x^6 + 14967*x^7 +...
A(A(x)) = x + 2*x^2 + 12*x^3 + 102*x^4 + 1042*x^5 + 11977*x^6 + 149776*x^7 +...
A(A(x))^3/A(x) = x^2 + 5*x^3 + 38*x^4 + 357*x^5 + 3832*x^6 + 45189*x^7 + ...
The g.f. satisfies:
A(x) = x + A(x)^3/x + [d/dx A(x)^6/x^2]/2! + [d^2/dx^2 A(x)^9/x^3]/3! + [d^3/dx^3 A(x)^12/x^4]/4! +...
Logarithmic series:
log(A(x)/x) = A(x)^3/x^2 + [d/dx A(x)^6/x^3]/2! + [d^2/dx^2 A(x)^9/x^4]/3! + [d^3/dx^3 A(x)^12/x^5]/4! +...
From _Seiichi Manyama_, Jun 05 2025: (Start)
Let b(n,k) = [x^n] (A(x)/x)^k.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,3*j).
a(n) = b(n-1,1). (End)
		

Crossrefs

Cf. A213591.

Programs

PARI

{a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^3/x+x*O(x^n))); polcoeff(A, n))}

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(3*m)/x^m)/m!)+x*O(x^n)); polcoeff(A, n)}

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(3*m)/x^(m+1))/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 21, print1(a(n), ", "))

PARI

b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 3*j)));
a(n) = b(n-1, 1); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies:

(1) A(x) = x + A(A(x))^3 / A(x).

(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x^n / n!.

(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x^(n+1) / n! ).