This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370442 #12 Feb 18 2024 14:52:16
%S A370442 1,2,12,78,570,4434,36174,305142,2640612,23311068,209111736,
%T A370442 1900666896,17466522690,162014855658,1514885838582,14263411673472,
%U A370442 135117683341050,1286880634334490,12315286940334942,118362499698060384,1141990331203349562,11056838563337857548,107394670044059002968
%N A370442 Expansion of g.f. A(q) satisfying -2 = Product_{n>=0} (1 - 3*q^n*A(q)).
%H A370442 Paul D. Hanna, <a href="/A370442/b370442.txt">Table of n, a(n) for n = 0..400</a>
%F A370442 G.f. A(q) = Sum_{n>=0} a(n)*q^n satisfies the following formulas.
%F A370442 (1) -2 = Product_{n>=0} (1 - 3*q^n*A(q)).
%F A370442 (2) -2 = Sum_{n>=0} (-3)^n * q^(n*(n-1)/2) * A(q)^n / Product_{k=1..n} (1 - q^k).
%F A370442 (3) A(q) = lim_{n->oo} R_{n}(q) / R_{n+1}(q) where R_{n}(q) = faq(n,q) * [x^n] 3/(1 + 2*e_q(3*x,q)), e_q(x,q) is the q-exponential of x and faq(n,q) is the q-factorial of n.
%F A370442 a(n) ~ c * d^n / n^(3/2), where d = 10.3914965886269147720490605009350781702243358825286425537327254915874... and c = 0.49970395101356434785108820969954986510927554236884857759717688447784... - _Vaclav Kotesovec_, Feb 18 2024
%e A370442 G.f.: A(q) = 1 + 2*q + 12*q^2 + 78*q^3 + 570*q^4 + 4434*q^5 + 36174*q^6 + 305142*q^7 + 2640612*q^8 + 23311068*q^9 + 209111736*q^10 + ...
%e A370442 where A(q) satisfies the infinite product
%e A370442 -2 = (1 - 3*A(q)) * (1 - 3*q*A(q)) * (1 - 3*q^2*A(q)) * (1 - 3*q^3*A(q)) * (1 - 3*q^4*A(q)) * (1 - 3*q^5*A(q)) * ...
%t A370442 (* Calculation of constants {d,c}: *) With[{m = 3}, {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 A370442 (PARI) /* A(q) satisfies -2 = Product_{n>=0} (1 - 3*q^n*A(q)) */
%o A370442 {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A370442 A[#A] = polcoeff( 2 + prod(k=0,#A, 1 - 3*x^k*Ser(A)) /3, #A-1, x) ); H=A; A[n+1]}
%o A370442 for(n=0,30, print1(a(n),", "))
%o A370442 (PARI) /* limit_{n->oo} R_{n}(q) / R_{n+1}(q) */
%o A370442 {faq(n,q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n
%o A370442 {R(n) = faq(n,q) * polcoeff( 3/(1 + 2*sum(m=0, n, (3*x)^m/faq(m,q) + x*O(x^(n+2)))), n, x)}
%o A370442 {a(n) = polcoeff(R(n+2)/R(n+3) + q*O(q^n), n, q)}
%o A370442 for(n=0,30, print1(a(n),", "))
%Y A370442 Cf. A152807, A370443, A370444, A370445.
%K A370442 nonn
%O A370442 0,2
%A A370442 _Paul D. Hanna_, Feb 18 2024