A357791 - OEIS

%I #8 Jan 11 2023 10:22:54

%S 1,1,2,5,21,88,377,1654,7424,34000,158274,746525,3559456,17128250,

%T 83078147,405754479,1993777057,9849668910,48892589632,243739139810,

%U 1219789105228,6125813250402,30862120708266,155937956267432,790019313067409,4012282344217699,20423575546661000

%N a(n) = coefficient of x^n in A(x) such that: x = Sum_{n=-oo..+oo} x^n * (1 - x^n *  A(-x)^n)^n.

%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

%H Paul D. Hanna, <a href="/A357791/b357791.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.

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

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

%F a(n) ~ c * d^n / n^(3/2), where d = 5.390297559554269719991046... and c = 0.267652299887938085649... - _Vaclav Kotesovec_, Dec 25 2022

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 21*x^4 + 88*x^5 + 377*x^6 + 1654*x^7 + 7424*x^8 + 34000*x^9 + 158274*x^10 + 746525*x^11 + 3559456*x^12 + ...

%e SPECIFIC VALUES.

%e A(x) = 3/2 at x = 0.1850570503493984408934312903280642188437354418734...

%e A(1/6) = 1.3085832721715442420948608003299892250459754159045...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);

%o A[#A] = polcoeff(x + sum(n=-#A, #A, (-x)^n * (1 - (-x)^n * Ser(A)^n )^n ), #A-1) ); A[n+1]}

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

%Y Cf. A357399, A359672.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 24 2022