|
%I #8 May 25 2023 08:58:26
%S 1,2,6,20,60,196,644,2128,7178,24374,83496,288420,1002272,3503748,
%T 12311818,43458316,154038006,548018604,1956263020,7004845080,
%U 25153186956,90554989440,326790211458,1181910952584,4283416505940,15553332981066,56575492155764,206136324338908
%N Expansion of g.f. A(x) satisfying 2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^(n-2))^(n+1).
%H Paul D. Hanna, <a href="/A363107/b363107.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
%F (1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^(n-2))^(n+1).
%F (2) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(4*n) * (2*A(x) + x^(n-1))^n.
%F (3) 2*x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+2))^(n-1).
%F (4) 2*x^2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+2))^(n+1).
%F (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^(n-2))^n.
%F (6) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(3*n-2) * (2*A(x) + x^(n-2))^(n-1).
%F (7) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^(n+2))^(n+1).
%F (8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^(n-1))^n.
%F (9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+2))^n.
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 60*x^4 + 196*x^5 + 644*x^6 + 2128*x^7 + 7178*x^8 + 24374*x^9 + 83496*x^10 + 288420*x^11 + 1002272*x^12 + ...
%o (PARI) {a(n) = my(A=[1], y=2); for(i=1, n, A = concat(A, 0);
%o A[#A] = polcoeff(y - sum(n=-#A, #A, (-1)^n * x^(2*n) * (y*Ser(A) + x^(n-2))^(n+1) )/y, #A-1, x) ); A[n+1]}
%o for(n=0, 30, print1( a(n), ", "))
%o (PARI) {a(n) = my(A=1, y=2); for(i=1, n,
%o A = 1/sum(m=-n,n, (-1)^m * x^(2*m) * (y*A + x^(m-2) + x*O(x^n) )^m ) );
%o polcoeff( A, n, x)}
%o for(n=0, 30, print1( a(n), ", "))
%Y Cf. A359712, A363106, A363108, A363109.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 24 2023
| 