cp's OEIS Frontend

A359713 a(n) = coefficient of x^n in A(x) such that 3 = Sum_{n=-oo..+oo} (-x)^n * (3*A(x) + x^(n-1))^(n+1).

%I A359713 #12 May 22 2023 02:15:30
%S A359713 1,5,31,206,1433,10329,76459,577855,4440538,34591555,272545144,
%T A359713 2168118299,17390330046,140486973983,1142036572271,9335129425718,
%U A359713 76681549612006,632655728172281,5240339959916895,43561574812700958,363294379940353624,3038799803831856805
%N A359713 a(n) = coefficient of x^n in A(x) such that 3 = Sum_{n=-oo..+oo} (-x)^n * (3*A(x) + x^(n-1))^(n+1).
%H A359713 Paul D. Hanna, <a href="/A359713/b359713.txt">Table of n, a(n) for n = 0..200</a>
%F A359713 G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
%F A359713 (1) 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(n-1))^(n+1).
%F A359713 (2) 3*x = Sum_{n=-oo..+oo} (-1)^n * (3*x*A(x) + x^n)^(n+1).
%F A359713 (3) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 3*A(x)*x^(n+1))^(n-1).
%F A359713 (4) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * (3*x*A(x) + x^n)^n ].
%F A359713 (5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^(n+1))^n ].
%F A359713 a(n) = Sum_{k=0..n} A359670(n,k)*3^k for n >= 0.
%e A359713 G.f.: A(x) = 1 + 5*x + 31*x^2 + 206*x^3 + 1433*x^4 + 10329*x^5 + 76459*x^6 + 577855*x^7 + 4440538*x^8 + 34591555*x^9 + 272545144*x^10 + ...
%o A359713 (PARI) {a(n) = my(A=1,y=3); for(i=1,n,
%o A359713 A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
%o A359713 polcoeff( A,n,x)}
%o A359713 for(n=0,25, print1( a(n),", "))
%o A359713 (PARI) {a(n) = my(A=[1],y=3); for(i=1,n, A = concat(A,0);
%o A359713 A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
%o A359713 for(n=0,25, print1( a(n),", "))
%Y A359713 Cf. A359670, A359711, A359712, A363104, A363105.
%K A359713 nonn
%O A359713 0,2
%A A359713 _Paul D. Hanna_, Jan 17 2023