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A386648 E.g.f. A(x) satisfies 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.

%I A386648 #9 Aug 13 2025 10:17:33
%S A386648 1,1,10,63,806,9485,161540,2752155,59021506,1310350929,34127883032,
%T A386648 934203381287,28851653188814,942891341070237,33962521081521076,
%U A386648 1297690184864525043,53814377103189792794,2366017294084046632937,111607600081334119137488,5565256312162642805357247
%N A386648 E.g.f. A(x) satisfies 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.
%C A386648 Conjecture: for n > 6, a(n) (mod 6) equals [2,3,4,3,2,5] repeating.
%C A386648 In general, the following sums are equal:
%C A386648 (C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
%C A386648 (C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A386648 here, q = A(x) with p = LambertW(-x), r = 1.
%H A386648 Paul D. Hanna, <a href="/A386648/b386648.txt">Table of n, a(n) for n = 1..102</a>
%F A386648 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A386648 (1) 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.
%F A386648 (2) 1 = Sum_{n>=0} A(x)^(n^2) * exp( LambertW(-x) * A(x)^n ) / n!.
%F A386648 (3) 1 = Sum_{n>=0} A(x)^(n^2) * (-x/LambertW(-x))^(A(x)^n) / n!.
%F A386648 (4) 1 = Sum_{n>=0} A(x)^(n*(n+1))/n! * Sum_{k>=0} (A(x)^n - k)^(k-1) * (-x)^k/k!.
%e A386648 E.g.f.: A(x) = x + x^2/2! + 10*x^3/3! + 63*x^4/4! + 806*x^5/5! + 9485*x^6/6! + 161540*x^7/7! + 2752155*x^8/8! + 59021506*x^9/9! + 1310350929*x^10/10! + ...
%e A386648 where 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.
%e A386648 RELATED SERIES.
%e A386648 -LambertW(-x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + 6^5*x^6/6! + 7^6*x^7/7! + ... + n^(n-1)*x^n/n! + ...
%e A386648 where exp(LambertW(-x)) = -x/LambertW(-x);
%e A386648 also, (-x/LambertW(-x))^y = Sum_{k>=0} y*(y - k)^(k-1) * (-x)^k/k!.
%o A386648 (PARI) {a(n) = my(A=[0,1]); for(i=0, n, A=concat(A, 0);
%o A386648 A[#A] = polcoeff(1 - sum(m=0, #A, (Ser(A)^m + lambertw(-x +x^3*O(x^n)))^m /m! ), #A-1) ); n!*A[n+1]}
%o A386648 for(n=1, 20, print1(a(n), ", "))
%Y A386648 Cf. A386645.
%K A386648 nonn
%O A386648 1,3
%A A386648 _Paul D. Hanna_, Aug 12 2025