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E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} A(k*x)^(1/k).
1

%I #9 Sep 17 2023 21:27:56

%S 1,1,3,16,137,1806,36307,1120099,53543737,3987172576,462225323291,

%T 83073483132399,23029319036705797,9800155391415000808,

%U 6374883018317726171183,6314677980863430818696221,9492558648757349054617246833,21588387865629652184260447567056

%N E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} A(k*x)^(1/k).

%e G.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1806*x^5/5! + ...

%e where

%e A(x) = 1 + x*A(x) + x^2*A(x)*A(2*x)^(1/2)/2! + x^3*A(x)*A(2*x)^(1/2)*A(3*x)^(1/3)/3! + x^4*A(x)*A(2*x)^(1/2)*A(3*x)^(1/3)*A(4*x)^(1/4)/4! + ...

%e RELATED EXPANSIONS.

%e A(2*x)^(1/2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 825*x^4/4! + 22321*x^5/5! + ...

%e A(3*x)^(1/3) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2521*x^4/4! + 103606*x^5/5! + ...

%e A(4*x)^(1/4) = 1 + x + 9*x^2/2! + 169*x^3/3! + 5681*x^4/4! + 313521*x^5/5! + ...

%e A(5*x)^(1/5) = 1 + x + 11*x^2/2! + 256*x^3/3! + 10761*x^4/4! + 745726*x^5/5! + ...

%e A(6*x)^(1/6) = 1 + x + 13*x^2/2! + 361*x^3/3! + 18217*x^4/4! + 1519681*x^5/5! + ...

%e A(7*x)^(1/7) = 1 + x + 15*x^2/2! + 484*x^3/3! + 28505*x^4/4! + 2780646*x^5/5! + ...

%e A(8*x)^(1/8) = 1 + x + 17*x^2/2! + 625*x^3/3! + 42081*x^4/4! + 4699681*x^5/5! + ...

%e A(9*x)^(1/9) = 1 + x + 19*x^2/2! + 784*x^3/3! + 59401*x^4/4! + 7473646*x^5/5! + ...

%e A(10*x)^(1/10) = 1 + x + 21*x^2/2! + 961*x^3/3! + 80921*x^4/4! + 11325201*x^5/5! + ...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m/m!*prod(k=1, m, subst(A, x, k*x+x*O(x^n))^(1/k)))); n!*polcoeff(A, n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A230320, A230317.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 15 2013