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%I #13 Jan 17 2024 23:32:02
%S 1,2,6,32,190,1236,8482,60434,442788,3315046,25249888,195040914,
%T 1524256336,12030033178,95748941322,767655502862,6193902044684,
%U 50257335231264,409825115116030,3356850545246400,27606085924603602,227850606781308660,1886792409865105988
%N G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2).
%H Paul D. Hanna, <a href="/A354662/b354662.txt">Table of n, a(n) for n = 0..400</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F (1) 2 = Sum_{n=-oo..+oo} (-x)^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
%F (2) 2 = Sum_{n>=0} (-x)^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
%F (3) 2 = Sum_{n>=0} (-x)^(n*(n+1)/2) * (1 + A(x)^(2*n+1)) * A(x)^(n*(n-1)/2).
%F (4) 2 = Product_{n>=1} (1 - (-x)^n*A(x)^n) * (1 + (-x)^(n-1)*A(x)^n) * (1 + (-x)^n*A(x)^(n-1)), by the Jacobi triple product identity.
%F a(n) = (-1)^n * Sum_{k=0..2*n+1} A354649(n,k)*2^k, for n >= 0.
%F a(n) = -Sum_{k=0..2*n+1} A354650(n,k)*(-2)^k, for n >= 0.
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 32*x^3 + 190*x^4 + 1236*x^5 + 8482*x^6 + 60434*x^7 + 442788*x^8 + 3315046*x^9 + 25249888*x^10 + ...
%e such that A = A(x) satisfies:
%e (1) 2 = ... + x^36*A^28 + x^28*A^21 - x^21*A^15 - x^15*A^10 + x^10*A^6 + x^6*A^3 - x^3*A - x + 1 + A - x*A^3 - x^3*A^6 + x^6*A^10 + x^10*A^15 - x^15*A^21 - x^21*A^28 + x^28*A^36 +--+ ...
%e (2) 2 = (1-x) + (1-x^3)*A - x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 + x^10*(1-x^11)*A^15 - x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
%e (3) 2 = (1+A) - (1+A^3)*x - A*(1+A^5)*x^3 + A^3*(1+A^7)*x^6 + A^6*(1+A^9)*x^10 - A^10*(1+A^11)*x^15 - A^15*(1+A^13)*x^21 + A^21*(1+A^15)*x^28 + ...
%e (4) 2 = (1 + x*A)*(1 + A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 + x^2*A) * (1 + x^3*A^3)*(1 + x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 + x^4*A^3) * (1 + x^5*A^5)*(1 + x^4*A^5)*(1 - x^5*A^4) * ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A[#A] = -polcoeff(-2 + sum(m=0,sqrtint(2*#A+9), (-x)^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A354649, A354650, A354661, A354663, A354664, A268299, A354652, A354653, A354654.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 02 2022
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