A242004 - OEIS

%I #18 Nov 01 2019 11:52:31

%S 1,1,2,13,136,1901,32672,660213,15261866,396260409,11404802292,

%T 360239943502,12389377190088,460921028363253,18446977179761746,

%U 790450173217191235,36112553368752540450,1752531204026383726825,90044479726218309099544,4883556796657253767140501

%N G.f. satisfies: 2*A(x) = 1 + x + A(x*A(x)^2).

%H Vaclav Kotesovec, <a href="/A242004/b242004.txt">Table of n, a(n) for n = 0..320</a>

%F a(n) ~ c * 2^n * n^(n + 1/4*log(2)) / (exp(n) * (log(2))^n), where c = 0.52809869533428510...

%t nmax = 19; sol = {a[0] -> 1};

%t Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[2 A[x] - (1 + x + A[x A[x]^2]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];

%t sol /. Rule -> Set;

%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 01 2019 *)

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A = 1+x + subst(A,x,x*A^2 +x*O(x^n)) - A);polcoeff(A,n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A242003 (q=1), A242005 (q=3), A242006 (q=4).

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Aug 11 2014

%E Name corrected by _Vaclav Kotesovec_ and _Paul D. Hanna_, Aug 15 2014