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A337757 G.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * 4^n * ((1+x)^n - A(x))^n.

%I A337757 #11 Oct 13 2020 03:25:33
%S A337757 1,1,10,460,30250,2488776,240707480,26452491760,3233941091480,
%T A337757 433611348176880,63118887464611936,9899442124162104960,
%U A337757 1662993951689377716800,297806177944353392091200,56626969607275080551099520,11394470658417110387020266496,2419172929237326590857901776560,540511078482106447677809541679680
%N A337757 G.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * 4^n * ((1+x)^n - A(x))^n.
%C A337757 In general, the following sums are equal:
%C A337757 (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C A337757 (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
%C A337757 for any fixed integer k; here, k = 4 with r = 4, p = -A(x), q = (1+x).
%H A337757 Paul D. Hanna, <a href="/A337757/b337757.txt">Table of n, a(n) for n = 0..200</a>
%F A337757 G.f. A(x) satisfies:
%F A337757 (1) 1 = Sum_{n>=0} C(n+3,3) * 4^n * ( (1+x)^n - A(x) )^n.
%F A337757 (2) 1 = Sum_{n>=0} C(n+3,3) * 4^n * (1+x)^(n^2) / (1 + 4*(1+x)^n*A(x))^(n+4).
%F A337757 a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n! * n^(5/2), where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.0012636042138... - _Vaclav Kotesovec_, Oct 13 2020
%e A337757 G.f.: A(x) = 1 + x + 10*x^2 + 460*x^3 + 30250*x^4 + 2488776*x^5 + 240707480*x^6 + 26452491760*x^7 + 3233941091480*x^8 + ...
%e A337757 where
%e A337757 1 = 1  +  4*4*((1+x) - A(x))  +  10*4^2*((1+x)^2 - A(x))^2  +  20*4^3*((1+x)^3 - A(x))^3  +  35*4^4*((1+x)^4 - A(x))^4  +  56*4^5*((1+x)^5 - A(x))^5  +  84*4^6*((1+x)^6 - A(x))^6  +  120*4^7*((1+x)^7 - A(x))^7 + ... + C(n+3,3)*4^n*((1+x)^n - A(x))^n + ...
%e A337757 Also,
%e A337757 1 = 1/(1 + 4*A(x))^4  +  4*4*(1+x)/(1 + 4*(1+x)*A(x))^5  +  10*4^2*(1+x)^4/(1 + (1+x)^2*A(x))^6  +  20*4^3*(1+x)^9/(1 + 4*(1+x)^3*A(x))^7  +  35*4^4*(1+x)^16/(1 + 4*(1+x)^4*A(x))^8  +  56*4^5*(1+x)^25/(1 + 4*(1+x)^5*A(x))^9  +  84*4^6*(1+x)^36/(1 + 4*(1+x)^6*A(x))^10 + ... + C(n+3,3)*4^n*(1+x)^(n^2)/(1 + 4*(1+x)^n*A(x))^(n+4) + ...
%o A337757 (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)*(m+3)/3! * 4^m * ((1+x)^m - Ser(A))^m ) )[#A]/16 ); A[n+1]}
%o A337757 for(n=0, 30, print1(a(n), ", "))
%Y A337757 Cf. A303056, A337755, A337756.
%K A337757 nonn
%O A337757 0,3
%A A337757 _Paul D. Hanna_, Sep 18 2020