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 A145268 #38 Mar 02 2024 03:51:35
%S A145268 1,1,3,9,30,104,378,1414,5424,21208,84244,339008,1379173,5663078,
%T A145268 23439651,97692524,409650348,1727034770,7315915371,31124324364,
%U A145268 132926220818,569695276362,2449395461726,10561857055472,45664873651576
%N A145268 G.f. A(x) satisfies A(x) = 1/Product_{k>0} (1-x^k*A(x)).
%H A145268 Vaclav Kotesovec, <a href="/A145268/b145268.txt">Table of n, a(n) for n = 0..250</a>
%F A145268 G.f. A(x) satisfies:
%F A145268 (1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^n / Product_{k=1..n} (1-x^k) due to an identity of Euler. - _Paul D. Hanna_, May 21 2011
%F A145268 (2) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^n / [Product_{k=1..n} (1-x^k)*(1-x^k*A(x))] due to Cauchy's identity. - _Paul D. Hanna_, May 21 2011
%F A145268 (3) A(x) = 1 + Sum_{n>=1} x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)) due to an identity of Euler. - _Paul D. Hanna_, Feb 11 2012
%F A145268 (4) A(x) = exp( Sum_{n>=1} x^n*A(x)^n / (n*(1-x^n)) ). - _Paul D. Hanna_, Mar 16 2012
%F A145268 a(n) ~ c * d^n / n^(3/2), where d = 4.6001032462748928128832068474594... and c = 0.695157167276255862302452181... - _Vaclav Kotesovec_, Aug 12 2021
%F A145268 Radius of convergence r = 0.2173864251437807911560951549077... = 1/d and A(r) = 2.126717513863405832814236571639... satisfy (a) A(r) = 1 / Sum_{n>=1} r^n/(1 - r^n*A(r)) and (b) A(r) = 1 / Product_{n>=1} (1 - r^n*A(r)). - _Paul D. Hanna_, Mar 02 2024
%e A145268 G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 104*x^5 + 378*x^6 +...
%e A145268 The g.f. satisfies:
%e A145268 (0) A(x) = 1/((1-x*A(x)) * (1-x^2*A(x)) * (1-x^3*A(x)) *...).
%e A145268 (1) A(x) = 1 + x*A(x)/(1-x) + x^2*A(x)^2/((1-x)*(1-x^2)) + x^3*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) +...
%e A145268 (2) A(x) = 1 + x*A(x)/[(1-x)*(1-x*A(x))] + x^4*A(x)^2/[(1-x)*(1-x^2)*(1-x*A(x))*(1-x^2*A(x))] + x^9*A(x)^3/[(1-x)*(1-x^2)*(1-x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))] +...
%e A145268 (3) A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
%e A145268 (4) A(x) = exp( x*A(x)/(1-x) + x^2*A(x)^2/(2*(1-x^2)) + x^3*A(x)^3/(3*(1-x^3)) +...).
%t A145268 terms = 25; A[_] = 0; Do[A[x_] = 1/Product[1-x^k A[x], {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 15 2018 *)
%t A145268 (* Calculation of constants {d,c}: *) {1/r, Sqrt[(r*(s-1)*s^3*Derivative[0, 1][QPochhammer][s, r]) / (2*Pi*((s-1)^2 * (QPolyGamma[1, Log[s]/Log[r], r]/Log[r]^2) - s))]} /. FindRoot[{s*QPochhammer[s, r] == 1 - s, 1 + s/(1 - s) == (Log[1 - r] + QPolyGamma[0, Log[s]/Log[r], r])/Log[r]}, {r, 1/5}, {s, 2}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Sep 28 2023 *)
%o A145268 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1/prod(k=1,n,(1-x^k*A+x*O(x^n))));polcoeff(A,n)} /* _Paul D. Hanna_, May 21 2011 */
%o A145268 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*A^m/prod(k=1,m,(1-x^k+x*O(x^n)))));polcoeff(A,n)} /* _Paul D. Hanna_, May 21 2011 */
%o A145268 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,sqrtint(n+1),x^(m^2)*A^m/prod(k=1,m,(1-x^k)*(1-x^k*A+x*O(x^n)))));polcoeff(A,n)} /* _Paul D. Hanna_, May 21 2011 */
%o A145268 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)} /* _Paul D. Hanna_, Feb 11 2012 */
%o A145268 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1,n,x^m*A^m/(m*(1-x^m +x*O(x^n))))));polcoeff(A,n)} /* _Paul D. Hanna_, Mar 16 2012 */
%Y A145268 Cf. A210043, A145267, A196150.
%K A145268 easy,nonn
%O A145268 0,3
%A A145268 _Vladeta Jovovic_, Oct 05 2008
%E A145268 More terms from _Max Alekseyev_, Jan 31 2010