A145268 G.f. A(x) satisfies A(x) = 1/Product_{k>0} (1-x^k*A(x)).
1, 1, 3, 9, 30, 104, 378, 1414, 5424, 21208, 84244, 339008, 1379173, 5663078, 23439651, 97692524, 409650348, 1727034770, 7315915371, 31124324364, 132926220818, 569695276362, 2449395461726, 10561857055472, 45664873651576
Offset: 0
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 104*x^5 + 378*x^6 +... The g.f. satisfies: (0) A(x) = 1/((1-x*A(x)) * (1-x^2*A(x)) * (1-x^3*A(x)) *...). (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)) +... (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))] +... (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))) +... (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)) +...).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..250
Programs
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Mathematica
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 *) (* 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 *)
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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 */ -
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 */ -
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 */ -
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 */ -
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 */
Formula
G.f. A(x) satisfies:
(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
(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
(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
(4) A(x) = exp( Sum_{n>=1} x^n*A(x)^n / (n*(1-x^n)) ). - Paul D. Hanna, Mar 16 2012
a(n) ~ c * d^n / n^(3/2), where d = 4.6001032462748928128832068474594... and c = 0.695157167276255862302452181... - Vaclav Kotesovec, Aug 12 2021
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
Extensions
More terms from Max Alekseyev, Jan 31 2010