|
%I #8 Mar 30 2012 18:37:32
%S 1,1,2,4,9,21,50,122,303,763,1943,4996,12953,33824,88877,234824,
%T 623474,1662618,4451171,11959159,32235236,87145035,236226761,
%U 641942519,1748479813,4772529625,13052515077,35763350619,98158386548,269844628977,742940020480,2048366903124,5655092015428
%N G.f. satisfies: A(x) = Sum_{n>=0} A(x)^n * x^(n*(n+1)/2) * (1 - x^(n+1))/(1 - x).
%F Define f(z,q) = Sum_{n>=0} z^n * q^(n*(n+1)/2) then g.f. A(q) satisfies:
%F A(q) = (f(A(q),q) - q*f(q*A(q),q))/(1-q).
%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 50*x^6 + 122*x^7 +...
%e where the g.f. satisfies the equivalent expressions:
%e A(x) = 1 + A(x)*x*(1-x^2)/(1-x) + A(x)^2*x^3*(1-x^3)/(1-x) + A(x)^3*x^6*(1-x^4)/(1-x) + A(x)^4*x^10*(1-x^5)/(1-x) +...
%e A(x) = 1 + A(x)*(x + x^2) + A(x)^2*(x^3 + x^4 + x^5) + A(x)^3*(x^6 + x^7 + x^8 + x^9) + A(x)^4*(x^10 + x^11 + x^12 + x^13 + x^14) +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,sqrtint(2*n+1),A^m*x^(m*(m+1)/2)*(1-x^(m+1))/(1-x))+x*O(x^n));polcoeff(A,n)}
%Y Cf. A199409.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 06 2011
|
