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A363509 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (3 + A(x^k)) * x^k/k ).

%I A363509 #15 May 10 2025 15:19:13
%S A363509 1,4,10,30,101,361,1354,5238,20740,83683,342719,1421019,5953306,
%T A363509 25162342,107163924,459438524,1981247950,8588054014,37398421941,
%U A363509 163534601567,717776072291,3161117717887,13964782042188,61866495037806,274792382789958
%N A363509 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (3 + A(x^k)) * x^k/k ).
%H A363509 Seiichi Manyama, <a href="/A363509/b363509.txt">Table of n, a(n) for n = 0..1000</a>
%F A363509 A(x) = Sum_{k>=0} a(k) * x^k = (1+x)^3 * Product_{k>=0} (1+x^(k+1))^a(k).
%F A363509 a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( 3 * (-1)^k + Sum_{d|k} (-1)^(k/d) * d * a(d-1) ) * a(n-k).
%t A363509 terms = 25; A[_] = 0; Do[A[x_] = Exp[Sum[(-1)^(k+1)*(3+A[x^k])*x^k/k,{k,terms}]]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Stefano Spezia_, May 10 2025 *)
%o A363509 (PARI) seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (-1)^(k+1)*(3+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);
%Y A363509 Cf. A004111, A038075, A038076, A363510.
%Y A363509 Cf. A363507.
%K A363509 nonn
%O A363509 0,2
%A A363509 _Seiichi Manyama_, Jun 06 2023