cp's OEIS Frontend

A258903 E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(3*x^n) / n!.

%I A258903 #12 Aug 09 2025 09:56:04
%S A258903 1,6,30,78,426,582,12450,4758,407010,2218182,19172370,360438,
%T A258903 4755166050,3213222,85631151090,5099958831318,54483404779650,
%U A258903 258673542,11939347971403410,2326095798,5556296851712151330,35398724239897109862,10235928407592878130,188311523478,758680053859872239555010
%N A258903 E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(3*x^n) / n!.
%C A258903 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 7 is [6, 2, 1, 6, 1, 4, 5, 2, 1, 0, 1, 5, 5, 2, 1, 0, 1, 5, 5, 2, 1, 0, 1, 5, 5, 2, 1, 0, 1, 5, ...] with an apparent period of 6 (= phi(7)) starting at n = 7. - _Peter Bala_, Aug 08 2025
%H A258903 Paul D. Hanna, <a href="/A258903/b258903.txt">Table of n, a(n) for n = 0..300</a>
%F A258903 E.g.f.: 2 - exp(3) + Sum_{n>=1} 3^n * exp(2*x^n) / n!.
%F A258903 For n >= 1, a(n) = Sum_{d divides n} 2^d * 3^(n/d) * n!/(d!*(n/d)!). - _Peter Bala_, Aug 08 2025
%e A258903 E.g.f.: A(x) = 1 + 6*x + 30*x^2/2! + 78*x^3/3! + 426*x^4/4! + 582*x^5/5! +...
%e A258903 where
%e A258903 A(x) = 2 - exp(2) + 2*exp(3*x) + 2^2*exp(3*x^2)/2! + 2^3*exp(3*x^3)/3! + 2^4*exp(3*x^4)/4! + 2^5*exp(3*x^5)/5! +...
%e A258903 A(x) = 2 - exp(3) + 3*exp(2*x) + 3^2*exp(2*x^2)/2! + 3^3*exp(2*x^3)/3! + 3^4*exp(2*x^4)/4! + 3^5*exp(2*x^5)/5! +...
%p A258903 with(numtheory): seq(n!*add(2^d*3^(n/d)/(d!*(n/d)!), d in divisors(n)), n = 1..25); # _Peter Bala_, Aug 08 2025
%o A258903 (PARI) {a(n) = local(A=1); A = 2-exp(2) + sum(m=1,n,2^m/m!*exp(3*x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
%o A258903 for(n=0,30, print1(a(n),", "))
%o A258903 (PARI) {a(n) = local(A=1); A = 2-exp(3) + sum(m=1,n,3^m/m!*exp(2*x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
%o A258903 for(n=0,30, print1(a(n),", "))
%Y A258903  Cf. A258899.
%K A258903 nonn
%O A258903 0,2
%A A258903 _Paul D. Hanna_, Jun 20 2015