cp's OEIS Frontend

A359672 a(n) = coefficient of x^n in A(x) where x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n.

%I A359672 #16 Jan 11 2023 11:09:33
%S A359672 1,1,2,5,21,72,257,998,3988,16064,65734,273541,1151184,4886946,
%T A359672 20916523,90181047,391230537,1706503782,7480000600,32930469730,
%U A359672 145546039760,645574246834,2872745389578,12821285282360,57377599801569,257416078950987,1157519956026736,5216112572700566
%N A359672 a(n) = coefficient of x^n in A(x) where x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n.
%H A359672 Paul D. Hanna, <a href="/A359672/b359672.txt">Table of n, a(n) for n = 0..300</a>
%F A359672 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A359672 (1) x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n.
%F A359672 (2) x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(n*(n-1)) * A(x)^(n^2) / (1 + x^n*A(x)^n)^n.
%F A359672 a(n) ~ c * d^n / n^(3/2), where d = 4.76347639696677679... and c = 0.37393658540119283... - _Vaclav Kotesovec_, Jan 11 2023
%e A359672 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 21*x^4 + 72*x^5 + 257*x^6 + 998*x^7 + 3988*x^8 + 16064*x^9 + 65734*x^10 + 273541*x^11 + 1151184*x^12 + ...
%e A359672 where
%e A359672 x = ... + x^6*A(x)^9/(1 + x^3*A(x)^3)^3 - x^2*A(x)^4/(1 + x^2*A(x)^2)^2 + A(x)/(1 + x*A(x)) - 1 + x*(1 + x*A(x)) - x^2*(1 + x^2*A(x)^2)^2 + x^3*(1 + x^3*A(x)^3)^3 + ... + (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n + ...
%e A359672 SPECIFIC VALUES.
%e A359672 A(1/d) = 1.71831164... where d = 4.76347639696677679... is given in the formula section.
%e A359672 A(1/5) = 1.47621312973364884841150188176844829427560286588046...
%o A359672 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A359672 A[#A] = polcoeff(x - sum(m=-#A,#A, (-1)^(m-1) * x^m * (1 + (x*Ser(A))^m)^m ), #A-1));A[n+1]}
%o A359672 for(n=0,30,print1(a(n),", "))
%Y A359672 Cf. A357791, A357399, A357797.
%K A359672 nonn
%O A359672 0,3
%A A359672 _Paul D. Hanna_, Jan 10 2023