A322725 - OEIS

%I #4 Jan 30 2019 17:46:56

%S 1,2,5,14,50,220,1107,6030,34643,207704,1293190,8332942,55406884,

%T 379151494,2664359328,19193917324,141571867121,1068156408852,

%U 8238449274801,64921172524532,522489723684089,4293039694194962,36000331681298631,308011504511924924,2687885268655409430,23916543285143972648,216912090405180557549

%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (2 + x*A(x)^n)^n.

%F G.f. A(x) satisfies:

%F (1) A(x) = Sum_{n>=0} x^n * (2 + x*A(x)^n)^n.

%F (2) A(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2) / (1 - 2*x*A(x)^n)^(n+1).

%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 50*x^4 + 220*x^5 + 1107*x^6 + 6030*x^7 + 34643*x^8 + 207704*x^9 + 1293190*x^10 + ...

%e such that

%e A(x) = 1 + x*(2 + x*A(x)) + x^2*(2 + x*A(x)^2)^2 + x^3*(2 + x*A(x)^3)^3 + x^4*(2 + x*A(x)^4)^4 + x^5*(2 + x*A(x)^5)^5 + x^6*(2 + x*A(x)^6)^6 + ...

%e Also, the g.f. satisfies the identity:

%e A(x) = 1/(1 - 2*x) + x^2*A(x)/(1 - 2*x*A(x))^2 + x^4*A(x)^4/(1 - 2*x*A(x)^2)^3 + x^6*A(x)^9/(1 - 2*x*A(x)^3)^4 + x^8*A(x)^16/(1 - 2*x*A(x)^4)^5 + x^10*A(x)^25/(1 - 2*x*A(x)^5)^6 + ...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*(2 + x*(A+x*O(x^n))^m)^m)); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, x^(2*k)*A^(k^2)/(1 - 2*x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A186998, A203014, A300049.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 30 2019