A192317 G.f.: A(x) = Sum_{n>=0} x^n/(1-x)^A038722(n), where A038722(n) = floor(sqrt(2*n)+1/2)^2 - n + 1.

1, 1, 2, 5, 10, 21, 47, 103, 217, 451, 951, 2047, 4439, 9548, 20231, 42355, 88373, 185343, 392297, 836502, 1787158, 3803651, 8035998, 16846041, 35121641, 73103052, 152493454, 319600236, 673256721, 1423293503, 3011396839, 6358756643, 13372146841

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).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 21*x^5 + 47*x^6 +...
which satisfies:
A(x) = 1 + x/(1-x) + x^2/(1-x)^3 + x^3/(1-x)^2 + x^4/(1-x)^6 + x^5/(1-x)^5 + x^6*(1+x)^4 +...
A(x) = 1 + (x/(1-x)) + (x/(1-x))^2*(1/(1-x)^2-x^2)/(1/(1-x)-x) + (x/(1-x))^4*(1/(1-x)^3-x^3)/(1/(1-x)-x) + (x/(1-x))^7*(1/(1-x)^4-x^4)/(1/(1-x)-x) + (x/(1-x))^11*(1/(1-x)^5-x^5)/(1/(1-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. A192316, A038722.

Programs

PARI

{a(n)=polcoeff(1+sum(m=1,sqrtint(2*n)+2,(x/(1-x+x*O(x^n)))^(m*(m-1)/2+1)/(1-x)^(m-1)*(1-x^m*(1-x)^m)/(1-x*(1-x))),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);A=sum(m=0,n,x^m/(1-x+x*O(x^n))^A038722(m));polcoeff(A,n)}

Formula

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

G.f.: A(x) = Sum_{n>=0} x^A038722(n)/(1-x)^n.