cp's OEIS Frontend

A258899 E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(x^n) / n!.

%I A258899 #17 Aug 09 2025 09:56:00
%S A258899 1,2,6,10,42,34,786,130,17058,81154,545346,2050,102457218,8194,
%T A258899 1141636866,72648608770,648648065538,131074,111258180895746,524290,
%U A258899 40892974286411778,229774078552113154,28890711351291906,8388610,3552178288049960329218,34469355651846669074434
%N A258899 E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(x^n) / n!.
%C A258899 Conjecture: the sequence a(n) taken modulo a positive integer k is eventually periodic with the period dividing phi(k). For example, the sequence taken modulo 11 is [1, 2, 6, 10, 9, 1, 5, 9, 8, 7, 10, 4, 6, 10, 7, 1, 0, 9, 5, 8, 3, 4, 6, 10, 7, 1, 0, 9, 5, 8, 3, 4, 6, 10, 7, 1, 0, 9, 5, 8, 3, ...] with an apparent period of 10 (= phi(11)) starting at n = 11. - _Peter Bala_, Aug 03 2025
%H A258899 Paul D. Hanna, <a href="/A258899/b258899.txt">Table of n, a(n) for n = 0..500</a>
%F A258899 E.g.f.: 2 - exp(1) + Sum_{n>=1} exp(2*x^n) / n!.
%F A258899 For n >= 1, a(n) = Sum_{d divides n} 2^d * n!/(d!*(n/d)!). - _Peter Bala_, Aug 04 2025
%e A258899 E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 10*x^3/3! + 42*x^4/4! + 34*x^5/5! + 786*x^6/6! +...
%e A258899 where
%e A258899 A(x) = 2 - exp(2) + 2*exp(x) + 2^2*exp(x^2)/2! + 2^3*exp(x^3)/3! + 2^4*exp(x^4)/4! + 2^5*exp(x^5)/5! +...
%e A258899 A(x) = 2 - exp(1) + exp(2*x) + exp(2*x^2)/2! + exp(2*x^3)/3! + exp(2*x^4)/4! + exp(2*x^5)/5! +...
%p A258899 with(numtheory): seq(`if`(n=0, 1, n!*add(2^d/(d!*(n/d)!), d in divisors(n))), n = 0..25); # _Peter Bala_, Aug 04 2025
%o A258899 (PARI) {a(n) = local(A=1); A = 2-exp(2) + sum(m=1,n,2^m/m!*exp(x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
%o A258899 for(n=0,30, print1(a(n),", "))
%o A258899 (PARI) {a(n) = local(A=1); A = 2-exp(1) + sum(m=1,n,1/m!*exp(2*x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
%o A258899 for(n=0,30, print1(a(n),", "))
%Y A258899 Cf. A121860, A258903.
%K A258899 nonn
%O A258899 0,2
%A A258899 _Paul D. Hanna_, Jun 20 2015