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a(n) = coefficient of x^n, n >= 0, in A(x) such that: 2 = Sum_{n=-oo..+oo} x^(2*n) * (1 - x^n)^(2*n) * A(x)^n.
2

%I #8 Dec 03 2022 12:07:40

%S 1,2,4,6,12,18,52,92,278,606,1736,4378,11974,31826,86592,234514,

%T 645864,1763784,4904844,13503716,37762996,104922104,294474340,

%U 824706556,2322523264,6544002006,18497507308,52357115006,148540658418,421986604840,1201221586484

%N a(n) = coefficient of x^n, n >= 0, in A(x) such that: 2 = Sum_{n=-oo..+oo} x^(2*n) * (1 - x^n)^(2*n) * A(x)^n.

%C Related formal power series identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n)^n.

%H Paul D. Hanna, <a href="/A357546/b357546.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) 2 = Sum_{n=-oo..+oo} x^(2*n) * (1 - x^n)^(2*n) * A(x)^n.

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

%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 6*x^3 + 12*x^4 + 18*x^5 + 52*x^6 + 92*x^7 + 278*x^8 + 606*x^9 + 1736*x^10 + 4378*x^11 + 11974*x^12 + ...

%e such that A(x) satisfies the doubly infinite series

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

%e which can be split into infinite series P and Q given by

%e 2 = P + Q ;

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

%e Q = 1 + x^2*(1-x)^2*A(x) + x^4*(1-x^2)^4*A(x)^2 + x^6*(1-x^3)^6*A(x)^3 + ... + x^(2*n)*(1 - x^n)^(2*n)*A(x)^n + ...

%e Explicitly,

%e P = 1 - x^2 - 2*x^4 - 4*x^5 - 13*x^6 - 18*x^7 - 75*x^8 - 126*x^9 - 475*x^10 - 974*x^11 - 2991*x^12 - 7184*x^13 - 20257*x^14 - ...

%e Q = 1 + x^2 + 2*x^4 + 4*x^5 + 13*x^6 + 18*x^7 + 75*x^8 + 126*x^9 + 475*x^10 + 974*x^11 + 2991*x^12 + 7184*x^13 + 20257*x^14 + ...

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

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

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 17 2022