A210043 G.f. A(x) satisfies: A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^(n-1)).
1, 1, 2, 4, 10, 26, 73, 211, 629, 1912, 5913, 18531, 58739, 187963, 606416, 1970326, 6441623, 21175056, 69946082, 232054411, 772886274, 2583325555, 8662455004, 29132638803, 98240253058, 332105822674, 1125273780474, 3820859749502, 12999287203624
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Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 73*x^6 + 211*x^7 +... The g.f. satisfies the q-series identities: (0) A(x) = 1/( (1-x) * (1-x^2*A(x)) * (1-x^3*A(x)^2) * (1-x^4*A(x)^3) *...). (1) A(x) = 1 + x/(1-x*A(x)) + x^2/((1-x*A(x))*(1-x^2*A(x)^2)) + x^3/((1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)) +... (2) A(x) = 1 + x/(1-x) + x^2*A(x)/((1-x)*(1-x^2*A(x))) + x^3*A(x)^2/((1-x)*(1-x^2*A(x))*(1-x^3*A(x)^2)) +... (3) A(x) = 1 + x/((1-x)*(1-x*A(x))) + x^4*A(x)^2/((1-x)*(1-x^2*A(x))*(1-x*A(x))*(1-x^2*A(x)^2)) + x^9*A(x)^6/((1-x)*(1-x^2*A(x))*(1-x^3*A(x)^2)*(1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)) +... (4) A(x) = exp( x/(1-x*A(x)) + x^2/(2*(1-x^2*A(x)^2)) + x^3/(3*(1-x^3*A(x)^3)) +...).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..325 (terms 0..200 from Paul D. Hanna)
Programs
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Mathematica
nmax = 30; A[] = 0; Do[A[x] = 1/(1 - x)/Product[1 - x^k*A[x]^(k - 1), {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 28 2023 *) (* Calculation of constants {d,c}: *) {1/r, -s*Log[r*s]* Sqrt[(-1 + r*s)*(((-2 + s)*Log[r*s] + (-1 + s)*Log[1 - r*s] + (-1 + s)*QPolyGamma[0, Log[1/s]/Log[r*s], r*s])/ (2* Pi*(Log[r*s]*(4*r*(-1 + s)*s*ArcTanh[1 - 2*r*s] + 2*(-3 + s)*(-1 + r*s)*Log[r*s]^2 + (2 - 2*s + (-5 + 3*s)*(-1 + r*s)*Log[r*s])* Log[1 - r*s] + (-1 + s)*(-1 + r*s)*Log[1 - r*s]^2) + (-1 + r*s)* Log[r*s]*((-5 + 3*s)*Log[r*s] + 2*(-1 + s)*(1 + Log[1 - r*s]))* QPolyGamma[0, Log[1/s]/Log[r*s], r*s] + (-1 + s)*(-1 + r*s)*Log[r*s]* QPolyGamma[0, Log[1/s]/Log[r*s], r*s]^2 + (-1 + r*s)*((-1 + s)*(2*Log[1/s] + Log[r*s])* QPolyGamma[1, Log[1/s]/Log[r*s], r*s] + r*s*Log[r*s]^2*((-r)*s^3*Log[r*s]* Derivative[0, 2][QPochhammer][1/s, r*s] - 2*(-1 + s)* Derivative[0, 0, 1][QPolyGamma][0, Log[1/s]/Log[r*s], r*s])))))]} /. FindRoot[{s - 1 == s^2*QPochhammer[1/s, r*s], (s - 2)/s + ((s - 1)*(Log[1 - r*s] + QPolyGamma[0, Log[1/s]/Log[r*s], r*s]))/(s*Log[r*s]) + r*s^2*Derivative[0, 1][QPochhammer][1/s, r*s] == 0}, {r, 1/4}, {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^(k-1)+x*O(x^n)))); polcoeff(A, n)} -
PARI
{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^(m-1)/prod(k=1, m, (1-x^k*A^(k-1)+x*O(x^n))))); polcoeff(A, n)} -
PARI
{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m/prod(k=1, m, (1-x^k*A^k +x*O(x^n))))); polcoeff(A, n)} -
PARI
{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(m^2-m)/prod(k=1, m, (1-x^k*A^(k-1))*(1-x^k*A^k+x*O(x^n))))); polcoeff(A, n)} -
PARI
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m/m/(1-x^m*A^m +x*O(x^n))))); polcoeff(A, n)} for(n=0,35,print1(a(n),", "))
Formula
G.f. A(x) satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n / Product_{k=1..n} (1 - x^k*A(x)^k).
(2) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n-1) / Product_{k=1..n} (1 - x^k*A(x)^(k-1)).
(3) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^(n^2-n) / [Product_{k=1..n} (1 - x^k*A(x)^(k-1))*(1 - x^k*A(x)^k)].
(4) A(x) = exp( Sum_{n>=1} x^n/n / (1 - x^n*A(x)^n) ).
a(n) ~ c * d^n / n^(3/2), where d = 3.58867546756663411130633387... and c = 0.57644814981246742030509... - Vaclav Kotesovec, Aug 12 2021