A192318 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^n * A(x)^A038722(n), where A038722(n) = floor(sqrt(2*n)+1/2)^2 - n + 1.

1, 1, 2, 6, 18, 61, 218, 804, 3052, 11831, 46646, 186487, 754177, 3079767, 12681568, 52595999, 219515014, 921264092, 3885468897, 16459470468, 70001813240, 298785285316, 1279450906737, 5495145204550, 23665623371950, 102175095587827

Offset: 0

Author

Paul D. Hanna, Jun 27 2011

Keywords

• nonn

Comments

A038722 is a self-inverse permutation of the natural numbers. Thus, the function defined by g(x,y) = Sum_{n>=0} x^n*y^A038722(n) is symmetric: g(x,y) = g(y,x). What are the properties of a function A(x) that satisfies: A(x) = g(x,A(x)) = g(A(x),x)?

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 61*x^5 + 218*x^6 + 804*x^7 +...
which satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^2 + x^4*A(x)^6 + x^5*A(x)^5 + x^6*A(x)^4 +...
A(x) = 1 + x*A(x) + x^2*A(x)^2*(A(x)^2-x^2)/(A(x)-x) + x^4*A(x)^4*(A(x)^3-x^3)/(A(x)-x) + x^7*A(x)^7*(A(x)^4-x^4)/(A(x)-x) + x^11*A(x)^11*(A(x)^5-x^5)/(A(x)-x) +...
Sequence A038722 begins:
[1, 3,2, 6,5,4, 10,9,8,7, 15,14,13,12,11, 21,20,19,18,17,16, 28,27,...].
		

Crossrefs

Cf. A038722.

Programs

PARI

{b(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,sqrtint(2*n)+2,(x*A+x*O(x^n))^(m*(m-1)/2+1)*(A^m-x^m)/(A-x)));polcoeff(A,n)}

PARI

{A038722(n)=local(t=floor(1/2+sqrt(2*n))); if(n<1, 0, t^2-n+1)}
{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*(A+x*O(x^n))^A038722(m)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} (x*A(x))^(n*(n-1)/2+1) * (A(x)^n - x^n)/(A(x)-x).

G.f. satisfies: A(x) = Sum_{n>=0} x^A038722(n) * A(x)^n.