A192316 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, 4, 6, 9, 21, 35, 42, 70, 168, 330, 471, 561, 855, 1950, 4402, 8023, 11616, 14245, 18425, 33880, 78519, 172047, 320451, 500579, 668582, 819819, 1113658, 2046760, 4599060, 10174544, 20102845, 34677986, 52310993, 70115066, 87683799, 115847016

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 + 4*x^3 + 6*x^4 + 9*x^5 + 21*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+x^2) + (x+x^2)^2*((1+x)^2-x^2) + (x+x^2)^4*((1+x)^3-x^3) + (x+x^2)^7*((1+x)^4-x^4) + (x+x^2)^11*((1+x)^5-x^5) +...
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. A192317, A038722.

Programs

PARI

{a(n)=polcoeff(1+sum(m=1,sqrtint(2*n)+2,(x+x^2+x*O(x^n))^(m*(m-1)/2+1)*((1+x)^m-x^m)),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+x^2)^(n*(n-1)/2+1) * ((1+x)^n - x^n).

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