A302171 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x))^k.
1, 1, 4, 14, 54, 213, 880, 3724, 16143, 71227, 319067, 1447160, 6633530, 30682425, 143028870, 671293632, 3169572659, 15044993968, 71752624923, 343658572717, 1652266087698, 7971518032791, 38581202763318, 187269381724629, 911404238805468, 4446493502832481, 21742327471261176
Offset: 0
Keywords
Examples
G.f. A(x) = 1 + x + 4*x^2 + 14*x^3 + 54*x^4 + 213*x^5 + 880*x^6 + 3724*x^7 + 16143*x^8 + ... G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - x^2*A(x))^2 * (1 - x^3*A(x))^3 * ...).
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] = 1/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 = 5.177446537296361283814259811908762546749... and c = 0.81395777803098291048009263980507199... - Vaclav Kotesovec, Sep 27 2023
Radius of convergence r = 0.1931454033945844258723936803941781838... = 1/d and A(r) = 2.2252305561396523944672847657756264073... satisfy (1) A(r) = 1 / Sum_{n>=1} n*r^n/(1 - r^n*A(r)) and (2) A(r) = 1 / Product_{n>=1} (1 - r^n*A(r))^n. - Paul D. Hanna, Mar 02 2024