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A354664
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G.f. A(x) satisfies: 4 = Sum_{n=-oo..oo} (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2).
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9
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3, 28, 756, 28200, 1205228, 55731456, 2714642292, 137199520340, 7127794098792, 378292284479388, 20421818573265728, 1117886561607128940, 61904487399635790288, 3461693986652051482948, 195203095905903229325340, 11087371481682320212435332, 633751222047605882649272600
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OFFSET
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0,1
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 4 = Sum_{n=-oo..oo} (-x)^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) 4 = Sum_{n>=0} (-x)^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(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).
(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.
a(n) = (-1)^n * Sum_{k=0..2*n+1} A354649(n,k)*4^k, for n >= 0.
a(n) = -Sum_{k=0..2*n+1} A354650(n,k)*(-4)^k, for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 62.81220628370975097276726417958831026998790927499386157136003... and c = 0.71771306470564419436314253512374835316192083855385416486... - Vaclav Kotesovec, Jun 08 2022
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
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EXAMPLE
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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 + ...
such that A = A(x) satisfies:
(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 +--+ ...
(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 + ...
(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 + ...
(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) * ...
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MATHEMATICA
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(* 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 *)
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PROG
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(PARI) {a(n) = my(A=[3]); for(i=1, n, A = concat(A, 0);
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]}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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