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

1, 2, 6, 10, 42, 34, 786, 130, 17058, 81154, 545346, 2050, 102457218, 8194, 1141636866, 72648608770, 648648065538, 131074, 111258180895746, 524290, 40892974286411778, 229774078552113154, 28890711351291906, 8388610, 3552178288049960329218, 34469355651846669074434

Offset: 0

Author

Paul D. Hanna, Jun 20 2015

Keywords

• nonn

Comments

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

Examples

			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! +...
		

Links

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

Crossrefs

Cf. A121860, A258903.

Programs

Maple

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),", "))

Formula

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