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G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^(n+1) - A(x) )^n.
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%I #10 Oct 14 2020 08:00:02

%S 1,2,2,10,112,1670,30682,663606,16443254,458349374,14184612446,

%T 482476888374,17892738705864,718662489646314,31085968593760190,

%U 1441017859748316954,71281146361450601326,3748236082140499881942,208808936226479892694126,12286084218797404915838902,761413942238514103243322732,49577303456014047226843229946,3383829651598944830489407813422

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

%H Paul D. Hanna, <a href="/A304642/b304642.txt">Table of n, a(n) for n = 0..300</a>

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

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

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

%F a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.154769618099522133628... - _Vaclav Kotesovec_, Oct 14 2020

%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 10*x^3 + 112*x^4 + 1670*x^5 + 30682*x^6 + 663606*x^7 + 16443254*x^8 + 458349374*x^9 + 14184612446*x^10 + 482476888374*x^11 + ...

%e such that

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

%e Also,

%e 1 = 1/(1 + A(x)) + (1+x)^2/(1 + (1+x)*A(x))^2 + (1+x)^6/(1 + (1+x)^2*A(x))^3 + (1+x)^12/(1 + (1+x)^3*A(x))^4 + (1+x)^20/(1 + (1+x)^4*A(x))^5 + (1+x)^30/(1 + (1+x)^5*A(x))^6 + (1+x)^42/(1 + (1+x)^6*A(x))^7 + ...

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

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

%Y Cf. A303056, A304639.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 16 2018