A317340 - OEIS

%I #13 Aug 06 2018 06:45:57

%S 1,1,5,19,215,3221,60255,1328529,33525557,949932155,29820028655,

%T 1026798211345,38478492170097,1559272992752559,67959179777259427,

%U 3170787800382000429,157723858411070076821,8334021579932292731707,466249346920348934320021,27536331208734157654465035,1712202839967112789793314087,111816679758439520925160452089

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

%H Paul D. Hanna, <a href="/A317340/b317340.txt">Table of n, a(n) for n = 0..300</a>

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

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

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

%F a(n) ~ c * r^(2*n) * (1 + exp(1/r))^n * n^n / exp(n), where r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation (1 + exp(-1/r)) * LambertW(-exp(-1/r)/r) = -1/r, and c = 0.9243009841585606... - _Vaclav Kotesovec_, Aug 06 2018

%e G.f.: A(x) = 1 + x + 5*x^2 + 19*x^3 + 215*x^4 + 3221*x^5 + 60255*x^6 + 1328529*x^7 + 33525557*x^8 + 949932155*x^9 + 29820028655*x^10 + ...

%e such that

%e 1+x = 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+x = 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=[1,1]); 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. A304642, A303056.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 02 2018