A354664 - OEIS

%I #16 Jan 19 2024 08:01:04

%S 3,28,756,28200,1205228,55731456,2714642292,137199520340,

%T 7127794098792,378292284479388,20421818573265728,1117886561607128940,

%U 61904487399635790288,3461693986652051482948,195203095905903229325340,11087371481682320212435332,633751222047605882649272600

%N G.f. A(x) satisfies: 4 = Sum_{n=-oo..oo}  (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

%H Paul D. Hanna, <a href="/A354664/b354664.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) 4 = Sum_{n=-oo..oo}  (-x)^(n*(n-1)/2) * A(x)^(n*(n+1)/2).

%F (2) 4 = Sum_{n>=0}  (-x)^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).

%F (3) 4 = Sum_{n>=0}  (-1)^(n*(n+1)/2) * A(x)^(n*(n-1)/2) * (1 + A(x)^(2*n+1)) * x^(n*(n+1)/2).

%F (4) 4 = 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)*4^k, for n >= 0.

%F a(n) = -Sum_{k=0..2*n+1} A354650(n,k)*(-4)^k, for n >= 0.

%F a(n) ~ c * d^n / n^(3/2), where d = 62.81220628370975097276726417958831026998790927499386157136003... and c = 0.71771306470564419436314253512374835316192083855385416486... - _Vaclav Kotesovec_, Jun 08 2022

%F Formula (4) can be rewritten as the functional equation QPochhammer(-x*y) * QPochhammer(1/x, -x*y)/(1 - 1/x) * QPochhammer(-1/y, -x*y)/(1 + 1/y) = 4. - _Vaclav Kotesovec_, Jan 19 2024

%e G.f.: A(x) = 3 + 28*x + 756*x^2 + 28200*x^3 + 1205228*x^4 + 55731456*x^5 + 2714642292*x^6 + 137199520340*x^7 + 7127794098792*x^8 + ...

%e such that A = A(x) satisfies:

%e (1) 4 = ... + 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) 4 = (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) 4 = (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) 4 = (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) * ...

%t (* Calculation of constant d: *) 1/r /. FindRoot[{r*s * QPochhammer[1/r, -r*s] * QPochhammer[-1/s, -r*s] * QPochhammer[-r*s]/((-1 + r)*(1 + s)) == 4, -4*(Log[-r*s] - (1 + s)*QPolyGamma[0, 1, -r*s] + (1 + s) * QPolyGamma[0, -Log[-s]/Log[-r*s], -r*s]) / (s*Log[-r*s]) + 4*r*(1 + s) * Derivative[0, 1][QPochhammer][1/r, -r*s] / QPochhammer[1/r, -r*s] + r^2*s*QPochhammer[1/r, -r*s]*QPochhammer[-r*s] * Derivative[0, 1][QPochhammer][-1/s, -r*s]/(-1 + r) + 4*r*(1 + s)*Derivative[0, 1][QPochhammer][-r*s, -r*s] / QPochhammer[-r*s] == 0}, {r, 1/50}, {s, 2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 19 2024 *)

%o (PARI) {a(n) = my(A=[3]); for(i=1,n, A = concat(A,0);

%o A[#A] = -polcoeff(-4 + 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) );H=A;A[n+1]}

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

%Y Cf. A354649, A354650, A354661, A354662, A354663, A268299, A354652, A354653, A354654.

%K nonn

%O 0,1

%A _Paul D. Hanna_, Jun 02 2022