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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.

%I A192316 #9 Mar 30 2012 18:37:27
%S A192316 1,1,2,4,6,9,21,35,42,70,168,330,471,561,855,1950,4402,8023,11616,
%T A192316 14245,18425,33880,78519,172047,320451,500579,668582,819819,1113658,
%U A192316 2046760,4599060,10174544,20102845,34677986,52310993,70115066,87683799,115847016
%N 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.
%C A192316 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).
%F A192316 G.f.: A(x) = 1 + Sum_{n>=1} (x+x^2)^(n*(n-1)/2+1) * ((1+x)^n - x^n).
%F A192316 G.f.: A(x) = Sum_{n>=0} x^A038722(n) * (1+x)^n.
%e A192316 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 9*x^5 + 21*x^6 +...
%e A192316 which satisfies:
%e A192316 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 +...
%e A192316 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) +...
%e A192316 Sequence A038722 begins:
%e A192316 [1, 3,2, 6,5,4, 10,9,8,7, 15,14,13,12,11, 21,20,19,18,17,16, 28,27,...].
%o A192316 (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)}
%o A192316 (PARI) {A038722(n)=local(t=floor(1/2+sqrt(2*n))); if(n<1, 0, t^2-n+1)}
%o A192316 {a(n)=local(A=1+x);A=sum(m=0,n,x^m*(1+x+x*O(x^n))^A038722(m));polcoeff(A,n)}
%Y A192316 Cf. A192317, A038722.
%K A192316 nonn
%O A192316 0,3
%A A192316 _Paul D. Hanna_, Jun 27 2011