A363138 - OEIS

%I #7 Jun 13 2023 08:04:14

%S 1,2,4,10,32,110,380,1452,5444,21422,84348,339498,1375168,5638930,

%T 23273316,96829410,405119328,1704806800,7207449048,30607173180,

%U 130475372576,558195492452,2395675502344,10312175675736,44507875822784,192575428797954,835133872734696,3629372408642778

%N G.f. A(x) satisfies: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) - x^n)^n * (1 - x^n*A(x))^n.

%C Compare to identity: 0 = Sum_{n=-oo..+oo} x^(2*n) * (y - x^n)^n * (y + x^n)^n, which holds for all y.

%H Paul D. Hanna, <a href="/A363138/b363138.txt">Table of n, a(n) for n = 0..300</a>

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

%F (1) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) - x^n)^n * (1 - x^n*A(x))^n.

%F (2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / ( (A(x) - x^n)^n * (1 - x^n*A(x))^n ).

%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 32*x^4 + 110*x^5 + 380*x^6 + 1452*x^7 + 5444*x^8 + 21422*x^9 + 84348*x^10 + 339498*x^11 + ...

%e where A = A(x) satisfies

%e 0 = ... - x^12/((A - x^3)^3*(1 - x^3*A)^3) + x^4/((A - x^2)^2*(1 - x^2*A)^2) - 1/((A - x)*(1 - x*A)) + 1 - x^2*(A - x)*(1 - x*A) + x^4*(A - x^2)^2*(1 - x^2*A)^2 - x^6*(A - x^3)^3*(1 - x^3*A)^3 + ...

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

%o A[#A] = -polcoeff(sum(m=-#A, #A, (-1)^m * x^(2*m) * (Ser(A) - x^m)^m*(1 - x^m*Ser(A))^m ), #A-1)); A[n+1]}

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

%Y Cf. A354963.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 12 2023