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 A194042 #23 Jan 17 2024 09:43:19
%S A194042 1,8,92,1248,18590,294032,4848456,82433472,1434755717,25438412696,
%T A194042 457838000316,8342826818080,153615385821902,2853694780131056,
%U A194042 53419650732453288,1006670308475537216,19081647909610147674,363577167520213046504,6959630216092660689968
%N A194042 G.f. satisfies: A(x) = ( Sum_{n>=0} q^(n*(n+1)/2) )^8 where q=x*A(x).
%H A194042 Vaclav Kotesovec, <a href="/A194042/b194042.txt">Table of n, a(n) for n = 0..600</a>
%F A194042 Let q = x*A(x), then g.f. A(x) satisfies:
%F A194042 (1) A(x) = ( Sum_{n>=0} (2*n+1) * q^n/(1 - q^(2*n+1)) )^2,
%F A194042 (2) A(x) = Sum_{n>=0} (n+1)^3 * q^n/(1 - q^(2*n+2)),
%F A194042 (3) A(x) = Product_{n>=1} (1 + q^n)^8*(1 - q^(2*n))^8,
%F A194042 (4) A(x) = exp( Sum_{n>=1} 8*(q^n/(1 + q^n))/n ),
%F A194042 (5) A(x/F(x)^8) = F(x)^8 where F(x) = Sum_{n>=0} x^(n*(n+1)/2),
%F A194042 due to q-series identities.
%F A194042 a(n) ~ c * d^n / n^(3/2), where d = 20.757466132085824914671628173626682246438530051407107800521045715415164484946... and c = 1.12762897595401376103508613947431975160374771449653856610031074284717... - _Vaclav Kotesovec_, Oct 07 2020
%e A194042 G.f.: A(x) = 1 + 8*x + 92*x^2 + 1248*x^3 + 18590*x^4 + 294032*x^5 +...
%e A194042 where
%e A194042 (0) A(x)^(1/8) = 1 + x*A(x) + x^3*A(x)^3 + x^6*A(x)^6 + x^10*A(x)^10 + x^15*A(x)^15 + x^21*A(x)^21 +...
%e A194042 (1) A(x)^(1/2) = 1/(1-x*A(x)) + 3*x*A(x)/(1-x^3*A(x)^3) + 5*x^2*A(x)^2/(1-x^5*A(x)^5) + 7*x^3*A(x)^3/(1-x^7*A(x)^7) +...
%e A194042 (2) A(x) = 1/(1-x^2*A(x)^2) + 8*x*A(x)/(1-x^4*A(x)^4) + 27*x^2*A(x)^2/(1-x^6*A(x)^6) + 64*x^3*A(x)^3/(1-x^8*A(x)^8) +...
%e A194042 (3) A(x) = (1+x*A(x))^8*(1-x^2*A(x)^2)^8 * (1+x^2*A(x)^2)^8*(1-x^4*A(x)^4)^8 * (1+x^3*A(x)^3)^8*(1-x^6*A(x)^6)^8 * (1+x^4*A(x)^4)^8*(1-x^8*A(x)^8)^8 *...
%e A194042 (4) log(A(x)) = 8*x*A(x)/(1+x*A(x)) + 8*(x^2*A(x)^2/(1+x^2*A(x)^2))/2 + 8*(x^3*A(x)^3/(1+x^3*A(x)^3))/3 + 8*(x^4*A(x)^4/(1+x^4*A(x)^4))/4 +...
%e A194042 Related expansions begin:
%e A194042 _ A(x)^(1/8) = 1 + x + 8*x^2 + 93*x^3 + 1272*x^4 + 19058*x^5 + 302705*x^6 + 5007234*x^7 + 85341048*x^8 +...+ A194043(n)*x^n +...
%e A194042 _ A(x)^(1/2) = 1 + 4*x + 38*x^2 + 472*x^3 + 6685*x^4 + 102340*x^5 + 1649446*x^6 + 27574712*x^7 + 473750970*x^8 +...+ A194044(n)*x^n +...
%t A194042 (* Calculation of constant d: *) 1/r /. FindRoot[{2*s^(1/8) == QPochhammer[-1, r*s]*QPochhammer[r^2*s^2, r^2*s^2], r*s*Derivative[0, 1][QPochhammer][-1, r*s] / QPochhammer[-1, r*s] + r^2*s^(15/8) * QPochhammer[-1, r*s] * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] - 2*Log[1 - r^2*s^2]/Log[r^2*s^2] - 2*QPolyGamma[0, 1, r^2*s^2]/Log[r^2*s^2] == 1/8}, {r, 1/20}, {s, 2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 17 2024 *)
%o A194042 (PARI) {a(n)=local(A=1+x, T=sum(m=0, sqrtint(2*n+1), x^(m*(m+1)/2))+x*O(x^n)); A=(serreverse(x/T^8)/x); polcoeff(A, n)}
%o A194042 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,sqrt(2*n+1),(x*A+x*O(x^n))^(m*(m+1)/2))^8);polcoeff(A,n)}
%o A194042 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(2*m+1)*(x*A)^m/(1-(x*A+x*O(x^n))^(2*m+1)))^2);polcoeff(A,n)}
%o A194042 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^3*(x*A)^m/(1-(x*A+x*O(x^n))^(2*m+2))));polcoeff(A,n)}
%o A194042 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+(x*A)^m)*(1-(x*A)^(2*m)+x*O(x^n)))^8); polcoeff(A, n)}
%o A194042 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, 8*(x*A)^m/(1+(x*A)^m+x*O(x^n))/m))); polcoeff(A, n)}
%Y A194042 Cf. A106336, A194043, A194044.
%K A194042 nonn
%O A194042 0,2
%A A194042 _Paul D. Hanna_, Aug 12 2011