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A337756 G.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)*(n+2)/2! * 3^n * ((1+x)^n - A(x))^n.

%I A337756 #9 Oct 13 2020 03:15:00
%S A337756 1,1,6,180,7845,434448,28594494,2157238350,182404049175,
%T A337756 17026549342770,1735705779016158,191667825521201286,
%U A337756 22781050822806698709,2899308092950790588988,393385952195184523370994,56691647586489579559334352,8649001755741912766806347253,1392791055204268736953260163092
%N A337756 G.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)*(n+2)/2! * 3^n * ((1+x)^n - A(x))^n.
%C A337756 In general, the following sums are equal:
%C A337756 (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C A337756 (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
%C A337756 for any fixed integer k; here, k = 3 with r = 3, p = -A(x), q = (1+x).
%H A337756 Paul D. Hanna, <a href="/A337756/b337756.txt">Table of n, a(n) for n = 0..200</a>
%F A337756 G.f. A(x) satisfies:
%F A337756 (1) 1 = Sum_{n>=0} C(n+2,2) * 3^n * ( (1+x)^n - A(x) )^n.
%F A337756 (2) 1 = Sum_{n>=0} C(n+2,2) * 3^n * (1+x)^(n^2) / (1 + 3*(1+x)^n*A(x))^(n+3).
%F A337756 a(n) ~ c * (1 + 3*exp(1/r))^n * r^(2*n) * n! * n^(3/2), where r = 0.947093169766093813913446822751643203941993193936... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/3 and c = 0.00671991787239... - _Vaclav Kotesovec_, Oct 13 2020
%e A337756 G.f.: A(x) = 1 + x + 6*x^2 + 180*x^3 + 7845*x^4 + 434448*x^5 + 28594494*x^6 + 2157238350*x^7 + 182404049175*x^8 + ...
%e A337756 where
%e A337756 1 = 1  +  3*3*((1+x) - A(x))  +  6*3^2*((1+x)^2 - A(x))^2  +  10*3^3*((1+x)^3 - A(x))^3  +  15*3^4*((1+x)^4 - A(x))^4  +  21*3^5*((1+x)^5 - A(x))^5  +  28*3^6*((1+x)^6 - A(x))^6  +  38*3^7*((1+x)^7 - A(x))^7 + ... + C(n+2,2)*3^n*((1+x)^n - A(x))^n + ...
%e A337756 Also,
%e A337756 1 = 1/(1 + 3*A(x))^3  +  3*3*(1+x)/(1 + 3*(1+x)*A(x))^4  +  6*3^2*(1+x)^4/(1 + 3*(1+x)^2*A(x))^5  +  10*3^3*(1+x)^9/(1 + 3*(1+x)^3*A(x))^6  +  15*3^4*(1+x)^16/(1 + 3*(1+x)^4*A(x))^7  +  21*3^5*(1+x)^25/(1 + 3*(1+x)^5*A(x))^8  +  28*3^6*(1+x)^36/(1 + 3*(1+x)^6*A(x))^9 + ... + C(n+2,2)*3^n*(1+x)^(n^2)/(1 + 3*(1+x)^n*A(x))^(n+3) + ...
%o A337756 (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (m+1)*(m+2)/2! * 3^m * ((1+x)^m - Ser(A))^m ) )[#A]/9 ); A[n+1]}
%o A337756 for(n=0, 30, print1(a(n), ", "))
%Y A337756 Cf. A303056, A337755, A337757.
%K A337756 nonn
%O A337756 0,3
%A A337756 _Paul D. Hanna_, Sep 18 2020