A302287 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x))^k.
1, 1, 3, 10, 31, 102, 342, 1167, 4046, 14213, 50464, 180847, 653296, 2376406, 8697194, 32002219, 118322499, 439364380, 1637827543, 6126870808, 22993190147, 86542625565, 326607659370, 1235650643059, 4685502714403, 17804713119018, 67790202024365, 258579199501709, 988012193672223
Offset: 0
Keywords
Examples
G.f. A(x) = 1 + x + 3*x^2 + 10*x^3 + 31*x^4 + 102*x^5 + 342*x^6 + 1167*x^7 + 4046*x^8 + 14213*x^9 + 50464*x^10 + ... G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x))^2 * (1 + x^3*A(x))^3 * (1 + x^4*A(x))^4 * ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Programs
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Mathematica
nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x])^k, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)
Formula
a(n) ~ c * d^n / n^(3/2), where d = 4.01604513838270620496843653760987690323... and c = 2.07544072297996637757124624302382219... - Vaclav Kotesovec, Sep 27 2023
Radius of convergence r = 0.2490011853807768883971843288180859269 = 1/d and A(r) = 3.261386924996517219078267128734843819... satisfy (1) A(r) = 1 / Sum_{n>=1} n*r^n/(1 + r^n*A(r)) and (2) A(r) = Product_{n>=1} (1 + r^n*A(r))^n. - Paul D. Hanna, Mar 02 2024