A370444 Expansion of g.f. A(q) satisfying -4 = Product_{n>=0} (1 - 5*q^n*A(q)).

1, 4, 40, 460, 5940, 82060, 1188140, 17792060, 273299640, 4282438360, 68184114040, 1099949668960, 17940069922740, 295334808497460, 4900888478007740, 81893191113037760, 1376770060020516140, 23270650287508521860, 395214289798485048340, 6740892510015481994360, 115419341703097589417340

Offset: 0

Author

Paul D. Hanna, Feb 18 2024

Keywords

• nonn

Examples

			G.f.: A(q) = 1 + 4*q + 40*q^2 + 460*q^3 + 5940*q^4 + 82060*q^5 + 1188140*q^6 + 17792060*q^7 + 273299640*q^8 + 4282438360*q^9 + 68184114040*q^10 + ...
where A(q) satisfies the infinite product
-4 = (1 - 5*A(q)) * (1 - 5*q*A(q)) * (1 - 5*q^2*A(q)) * (1 - 5*q^3*A(q)) * (1 - 5*q^4*A(q)) * (1 - 5*q^5*A(q)) * ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A152807, A370442, A370443, A370445.

Programs

Mathematica

(* Calculation of constants {d,c}: *) With[{m = 5}, {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 *)

PARI

/* A(q) satisfies -4 = Product_{n>=0} (1 - 5*q^n*A(q)) */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( 4 + prod(k=0,#A, 1 - 5*x^k*Ser(A)) /5, #A-1, x) ); H=A; A[n+1]}
for(n=0,30, print1(a(n),", "))

PARI

/* limit_{n->oo} R_{n}(q) / R_{n+1}(q) */
{faq(n,q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n
{R(n) = faq(n,q) * polcoeff( 5/(1 + 4*sum(m=0, n, (5*x)^m/faq(m,q) + x*O(x^(n+2)))), n, x)}
{a(n) = polcoeff(R(n+1)/R(n+2) + q*O(q^n), n, q)}
for(n=0,30, print1(a(n),", "))

Formula

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

(1) -4 = Product_{n>=0} (1 - 5*q^n*A(q)).

(2) -4 = Sum_{n>=0} (-5)^n * q^(n*(n-1)/2) * A(q)^n / Product_{k=1..n} (1 - q^k).

(3) A(q) = lim_{n->oo} R_{n}(q) / R_{n+1}(q) where R_{n}(q) = faq(n,q) * [x^n] 5/(1 + 4*e_q(5*x,q)), e_q(x,q) is the q-exponential of x and faq(n,q) is the q-factorial of n.

a(n) ~ c * d^n / n^(3/2), where d = 18.4404204270365344730849662390340654154532966962429860615573702674131... and c = 0.52704660567512847547508143537941958515989557138934496501237493733513... - Vaclav Kotesovec, Feb 18 2024