A370443 - OEIS

%I #10 Feb 18 2024 14:52:03

%S 1,3,24,216,2184,23592,267144,3128472,37582680,460564632,5735093832,

%T 72359126376,923021734344,11884281689688,154243249784856,

%U 2015831498613720,26506024201097352,350404606655241768,4654501489433893512,62092356103141330584,831534637662059617368

%N Expansion of g.f. A(q) satisfying -3 = Product_{n>=0} (1 - 4*q^n*A(q)).

%H Paul D. Hanna, <a href="/A370443/b370443.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(q) = Sum_{n>=0} a(n)*q^n satisfies the following formulas.

%F (1) -3 = Product_{n>=0} (1 - 4*q^n*A(q)).

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

%F (3) A(q) = lim_{n->oo} R_{n}(q) / R_{n+1}(q) where R_{n}(q) = faq(n,q) * [x^n] 4/(1 + 3*e_q(4*x,q)), e_q(x,q) is the q-exponential of x and faq(n,q) is the q-factorial of n.

%F a(n) ~ c * d^n / n^(3/2), where d = 14.4231123176639449630408542507057843543532473958120624505916750124669... and c = 0.51707658317945675859732872615636765308571079799394608380170536716694... - _Vaclav Kotesovec_, Feb 18 2024

%e G.f.: A(q) = 1 + 3*q + 24*q^2 + 216*q^3 + 2184*q^4 + 23592*q^5 + 267144*q^6 + 3128472*q^7 + 37582680*q^8 + 460564632*q^9 + 5735093832*q^10 + ...

%e where A(q) satisfies the infinite product

%e -3 = (1 - 4*A(q)) * (1 - 4*q*A(q)) * (1 - 4*q^2*A(q)) * (1 - 4*q^3*A(q)) * (1 - 4*q^4*A(q)) * (1 - 4*q^5*A(q)) * ...

%t (* Calculation of constants {d,c}: *) With[{m = 4}, {1/r, -s*Log[r] * Sqrt[r*Derivative[0, 1][QPochhammer][m*s, r] / (2*Pi*(m - 1)*QPolyGamma[1, Log[m*s]/Log[r], r])]} /. FindRoot[{m + QPochhammer[m*s, r] == 1, Log[1 - r] + QPolyGamma[0, Log[m*s]/Log[r], r] == 0}, {r, 1/m^2}, {s, 2}, WorkingPrecision -> 70]] (* _Vaclav Kotesovec_, Feb 18 2024 *)

%o (PARI) /* A(q) satisfies -3 = Product_{n>=0} (1 - 4*q^n*A(q)) */

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

%o A[#A] = polcoeff( 3 + prod(k=0,#A, 1 - 4*x^k*Ser(A)) /4, #A-1, x) ); H=A; A[n+1]}

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

%o (PARI) /* limit_{n->oo} R_{n}(q) / R_{n+1}(q) */

%o {faq(n,q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n

%o {R(n) = faq(n,q) * polcoeff( 4/(1 + 3*sum(m=0, n, (4*x)^m/faq(m,q) + x*O(x^(n+2)))), n, x)}

%o {a(n) = polcoeff(R(n+1)/R(n+2) + q*O(q^n), n, q)}

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

%Y Cf. A152807, A370442, A370444, A370445.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 18 2024