A321602 - OEIS

%I #13 Oct 25 2020 13:00:49

%S 1,2,5,68,1521,45328,1660032,71548008,3533826841,196432984748,

%T 12128132342482,823366216285428,60966207548525287,4890600994792550264,

%U 422601696583826709492,39142599000082019249968,3869325702147169825040193,406650337650126697706078146,45281361448272561712508294157,5325916931170845646048163850556,659842223101960470758187538118437

%N G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(2*n) - A(x))^n.

%H Paul D. Hanna, <a href="/A321602/b321602.txt">Table of n, a(n) for n = 0..200</a>

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

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

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

%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 68*x^3 + 1521*x^4 + 45328*x^5 + 1660032*x^6 + 71548008*x^7 + 3533826841*x^8 + 196432984748*x^9 + 12128132342482*x^10 + ...

%e such that

%e 1 = 1  +  ((1+x)^2 - A(x))  +  ((1+x)^4 - A(x))^2  +  ((1+x)^6 - A(x))^3  +  ((1+x)^8 - A(x))^4  +  ((1+x)^10 - A(x))^5  +  ((1+x)^12 - A(x))^6  +  ((1+x)^14 - A(x))^7 + ...

%e Also,

%e 1 = 1/(1 + A(x))  +  (1+x)^2/(1 + (1+x)^2*A(x))^2  +  (1+x)^8/(1 + (1+x)^4*A(x))^3  +  (1+x)^18/(1 + (1+x)^6*A(x))^4  +  (1+x)^32/(1 + (1+x)^8*A(x))^5  +  (1+x)^50/(1 + (1+x)^10*A(x))^6  + ...

%e RELATED SERIES.

%e The logarithmic derivative of the g.f. begins

%e A'(x)/A(x) = 2 + 6*x + 182*x^2 + 5554*x^3 + 211172*x^4 + 9397920*x^5 + 476737830*x^6 + 27086036234*x^7 + 1702330030676*x^8 + ...

%e the coefficients of which are all even:

%e (1/2) * A'(x)/A(x) = 1 + 3*x + 91*x^2 + 2777*x^3 + 105586*x^4 + 4698960*x^5 + 238368915*x^6 + 13543018117*x^7 + 851165015338*x^8 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(2*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}

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

%Y Cf. A303056, A321603, A321604, A321605.

%Y Cf. A326262.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 14 2018