A337720 - OEIS

%I #11 Sep 20 2020 01:14:04

%S 1,1,1,1,2,4,8,22,63,190,618,2119,7617,28571,111271,448704,1869072,

%T 8024648,35445048,160820201,748525951,3569856592,17426990777,

%U 86999613910,443782312584,2311252408353,12281279591605,66538929813869,367350691232741,2065442073525419

%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1 + x*(1+x)^n)^n * x^n / A(x)^n.

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

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

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

%e G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + 22*x^7 + 63*x^8 + 190*x^9 + 618*x^10 + 2119*x^11 + 7617*x^12 + ...

%e where

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

%e also

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

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);

%o A[#A] = polcoeff( sum(m=0,#A, (1 + x*(1+x)^m +x*O(x^#A))^m * x^m / Ser(A)^(m+1)  ), #A-1));A[#A]}

%o for(n=0,30,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,#A, (1+x +x*O(x^#A))^(m^2) * x^(2*m) / ( Ser(A) - x*(1+x)^m )^(m+1) ), #A-1));A[#A]}

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

%Y Cf. A337721.

%K nonn

%O 0,5

%A _Paul D. Hanna_, Sep 16 2020