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%I #12 Mar 02 2024 13:11:07
%S 1,1,3,10,31,102,342,1167,4046,14213,50464,180847,653296,2376406,
%T 8697194,32002219,118322499,439364380,1637827543,6126870808,
%U 22993190147,86542625565,326607659370,1235650643059,4685502714403,17804713119018,67790202024365,258579199501709,988012193672223
%N G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x))^k.
%H Vaclav Kotesovec, <a href="/A302287/b302287.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 4.01604513838270620496843653760987690323... and c = 2.07544072297996637757124624302382219... - _Vaclav Kotesovec_, Sep 27 2023
%F 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
%e 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 + ...
%e 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 * ...
%t 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 *)
%Y Cf. A026007, A145267, A298260, A301456, A302171.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 04 2018
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