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A386657 E.g.f.: Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.

%I A386657 #8 Aug 24 2025 11:33:06
%S A386657 1,5,287,274532,4362420261,1131407873777920,4729288202285254702123,
%T A386657 317048074495318899943286044736,
%U A386657 340323907513179399929311813628104334217,5846207259092593125133941613189798019292422881280
%N A386657 E.g.f.: Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.
%C A386657 Conjecture: for n >= 6, a(n) (mod 6) equals [1, 2, 3, 2, 3, 4] repeating.
%C A386657 In general, the following sums are equal:
%C A386657 (C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
%C A386657 (C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A386657 here, q = 4 with p = LambertW(x)/x, r = x.
%F A386657 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A386657 (1) A(x) = Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.
%F A386657 (2) A(x) = Sum_{n>=0} 4^(n^2) * exp( LambertW(x) * 4^n ) * x^n / n!.
%F A386657 (3) A(x) = Sum_{n>=0} 4^(n^2) * (x/LambertW(x))^(4^n) * x^n / n!.
%F A386657 (4) A(x) = Sum_{n>=0} 4^(n*(n+1)) * x^n/n! * Sum_{k>=0} (4^n - k)^(k-1) * x^k/k!.
%F A386657 a(n) = Sum_{k=0..n} binomial(n,k) * 4^(k*(k+1)) * (4^k - (n-k))^(n-k-1).
%F A386657 a(n) = Sum_{k=0..n} binomial(n,k) * 4^(n*k) * (1 - (n-k)/4^k)^(n-k-1).
%e A386657 E.g.f.: A(x) = 1 + 5*x + 287*x^2/2! + 274532*x^3/3! + 4362420261*x^4/4! + 1131407873777920*x^5/5! + 4729288202285254702123*x^6/6! + ...
%e A386657 where A(x) = Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.
%e A386657 RELATED SERIES.
%e A386657 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! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
%e A386657 where exp(LambertW(x)) = x/LambertW(x);
%e A386657 also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.
%o A386657 (PARI) {a(n) = sum(k=0,n, binomial(n,k) * 4^(k*(k+1)) * (4^k - (n-k))^(n-k-1) )}
%o A386657 for(n=0, 12, print1(a(n), ", "))
%o A386657 (PARI) {a(n) = my(A = sum(m=0, n, (4^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
%o A386657 for(n=0, 12, print1(a(n), ", "))
%Y A386657 Cf. A386655 (q=2), A386656 (q=3), A386658 (q=5), A386648.
%K A386657 nonn,new
%O A386657 0,2
%A A386657 _Paul D. Hanna_, Aug 23 2025