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A386658 E.g.f.: Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.

%I A386658 #8 Aug 24 2025 11:32:57
%S A386658 1,6,674,2000229,153566609748,298500361403750381,
%T A386658 14557504055095871311168750,17765160070810827062009088144577731,
%U A386658 542112188572462226990932242595876785196798632,413592212104548192173492724488185195719396124921931347641
%N A386658 E.g.f.: Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.
%C A386658 Conjecture: for n >= 6, a(n) (mod 6) equals [4, 3, 0, 3, 0, 5] repeating.
%C A386658 In general, the following sums are equal:
%C A386658 (C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
%C A386658 (C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A386658 here, q = 5 with p = LambertW(x)/x, r = x.
%F A386658 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A386658 (1) A(x) = Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.
%F A386658 (2) A(x) = Sum_{n>=0} 5^(n^2) * exp( LambertW(x) * 5^n ) * x^n / n!.
%F A386658 (3) A(x) = Sum_{n>=0} 5^(n^2) * (x/LambertW(x))^(5^n) * x^n / n!.
%F A386658 (4) A(x) = Sum_{n>=0} 5^(n*(n+1)) * x^n/n! * Sum_{k>=0} (5^n - k)^(k-1) * x^k/k!.
%F A386658 a(n) = Sum_{k=0..n} binomial(n,k) * 5^(k*(k+1)) * (5^k - (n-k))^(n-k-1).
%F A386658 a(n) = Sum_{k=0..n} binomial(n,k) * 5^(n*k) * (1 - (n-k)/5^k)^(n-k-1).
%e A386658 E.g.f.: A(x) = 1 + 6*x + 674*x^2/2! + 2000229*x^3/3! + 153566609748*x^4/4! + 298500361403750381*x^5/5! + 14557504055095871311168750*x^6/6! + ...
%e A386658 where A(x) = Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.
%e A386658 RELATED SERIES.
%e A386658 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 A386658 where exp(LambertW(x)) = x/LambertW(x);
%e A386658 also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.
%o A386658 (PARI) {a(n,q=5) = sum(k=0,n, binomial(n,k) * q^(k*(k+1)) * (q^k - (n-k))^(n-k-1) )}
%o A386658 for(n=0, 12, print1(a(n), ", "))
%o A386658 (PARI) {a(n,q=5) = my(A = sum(m=0, n, (q^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
%o A386658 for(n=0, 12, print1(a(n), ", "))
%Y A386658 Cf. A386655 (q=2), A386656 (q=3), A386657 (q=4), A386648.
%K A386658 nonn,new
%O A386658 0,2
%A A386658 _Paul D. Hanna_, Aug 23 2025