|
%I #7 Apr 07 2022 12:13:48
%S 1,2,3,7,16,40,105,281,770,2147,6080,17431,50459,147265,432821,
%T 1279978,3806154,11373668,34136769,102863360,311065038,943740309,
%U 2871717550,8762164066,26801981013,82172139800,252470206882,777243596670,2397210487402,7406340364950
%N G.f. A(x) satisfies: 1 - x = Sum_{n>=0} x^n * (x^n + (-1)^n*A(x))^n.
%F G.f. A(x) satisfies:
%F (1) 1 - x = Sum_{n>=0} x^n * ( x^n + (-1)^n*A(x) )^n.
%F (2) 1 - x = Sum_{n>=0} x^(n*(n+1)) / ( 1 + (-1)^n*x^(n+1)*A(x) )^(n+1).
%e G.f.: A(x) = 1 + 2*x + 3*x^2 + 7*x^3 + 16*x^4 + 41*x^5 + 107*x^6 + 286*x^7 + 782*x^8 + 2179*x^9 + 6172*x^10 + 17702*x^11 + 51272*x^12 + ...
%e where
%e 1 - x = 1 + x*(x - A(x)) + x^2*(x^2 + A(x))^2 + x^3*(x^3 - A(x))^3 + x^4*(x^4 + A(x))^4 + x^5*(x^5 - A(x))^5 + x^6*(x^6 + A(x))^6 + ...
%e Also,
%e 1 - x = 1/(1 + x*A(x)) + x^2/(1 - x^2*A(x))^2 + x^6/(1 + x^3*A(x))^3 + x^12/(1 - x^4*A(x))^4 + x^20/(1 + x^5*A(x))^5 + ...
%e Specific values.
%e A(x) = 2 at x = 0.2515345800906597549860933755999098710430442043412...
%e A(x) = 3 at x = 0.3019496486281905495940467534583642873678976347045...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=0,#A, x^m * (x^m + (-1)^m*Ser(A))^m ),#A));A[n+1]}
%o for(n=0,40,print1(a(n),", "))
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=0,sqrtint(#A), x^((m+1)*m)/(1 + (-1)^m*x^(m+1)*Ser(A))^(m+1) ),#A));A[n+1]}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A352817, A317997.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 05 2022
|
