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Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^(n+1).
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%I #19 May 23 2023 07:12:55

%S 1,1,3,7,17,42,107,275,715,1884,5009,13421,36224,98382,268657,737244,

%T 2032035,5622938,15615186,43505382,121570407,340639265,956861955,

%U 2694064938,7601455079,21490621769,60870280259,172707869088,490818655346,1396973741672,3981748142925

%N Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^(n+1).

%H Paul D. Hanna, <a href="/A363142/b363142.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) 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^(n+1).

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

%F (3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^n.

%F (4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (A(x) + x^(2*n-1))^(n-1).

%F (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + A(x)*x^(2*n+1))^n.

%F a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) for n >= 0. - _Paul D. Hanna_, May 18 2023

%e G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 42*x^5 + 107*x^6 + 275*x^7 + 715*x^8 + 1884*x^9 + 5009*x^10 + 13421*x^11 + 36224*x^12 + ...

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

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

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

%Y Cf. A359711, A363143, A363144, A363182.

%Y Cf. A359670.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 17 2023