0,2

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

Paul D. Hanna, Table of n, a(n) for n = 0..500

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

For n >= 1, a(n) = Sum_{d divides n} 2^d * n!/(d!*(n/d)!). - Peter Bala, Aug 04 2025

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! +...

where

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! +...

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! +...

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

(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))}

for(n=0, 30, print1(a(n), ", "))

(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))}

for(n=0, 30, print1(a(n), ", "))

Cf. A121860, A258903.

Sequence in context: A387288 A202533 A275700 * A248784 A341337 A067868

Adjacent sequences: A258896 A258897 A258898 * A258900 A258901 A258902

Paul D. Hanna, Jun 20 2015

approved