A094717 a(n) = n! * Sum_{i+2j+3k=n} 1/(i!*(2j)!*(3k)!).

1, 1, 2, 5, 12, 36, 113, 351, 1080, 3281, 9882, 29646, 88817, 266085, 797526, 2391485, 7173360, 21520080, 64563521, 193700403, 581120892, 1743392201, 5230206126, 15690618378, 47071766561, 141215033961, 423644570442, 1270932914165, 3812797945332, 11438393835996

Offset: 0

Author

Benoit Cloitre, May 23 2004

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-6,12,-9).

Crossrefs

Cf. A024493, A049347, A057083, A057682, A094715, A099837.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-5*x+8*x^2-5*x^3+2*x^4-2*x^5)/((1-x)*(1-3*x)*(1+x+x^2)*(1-3*x+3*x^2)) )); // G. C. Greubel, Jul 14 2023

Maple

A094717_list := proc(n) local i; exp(z)*cosh(z)*(exp(z)+2*exp(-z/2)* cos(z*sqrt(3/4)))/3; series(%,z,n+2); seq(simplify(i!*coeff(%,z,i)),i=0..n) end: A094717_list(27); # Peter Luschny, Jul 11 2012

Mathematica

a[n_]:= n! Sum[Boole[i +2j +3k ==n]/(i! (2j)! (3k)!), {i,0,n}, {j,0,n}, {k,0,n}]; Table[a[n], {n,0,27}] (* Jean-François Alcover, Jul 06 2019 *)
LinearRecurrence[{6,-12,10,-6,12,-9}, {1,1,2,5,12,36}, 40] (* G. C. Greubel, Jul 14 2023 *)

PARI

a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,n,if(n-i-2*j-3*k,0,n!/(i)!/(2*j)!/(3*k)!))))

SageMath

def A094717_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x+8*x^2-5*x^3+2*x^4-2*x^5)/((1-x)*(1-3*x)*(1+x+x^2)*(1-3*x+3*x^2)) ).list()
A094717_list(40) # G. C. Greubel, Jul 14 2023

Formula

Limit_{n->oo} a(n)/3^n = 1/6.

E.g.f.: exp(z)*cosh(z)*(exp(z) + 2*exp(-z/2)*cos(z*sqrt(3/4)))/3. - Peter Luschny, Jul 11 2012

G.f.: (1-5*x+8*x^2-5*x^3+2*x^4-2*x^5)/((1-x)*(1-3*x)*(1+x+x^2)*(1-3*x+3*x^2)). - Colin Barker, Dec 24 2012

From G. C. Greubel, Jul 14 2023: (Start)

a(n) = (1/6)*(1 + 3^n + 2*A049347(n) + A049347(n-1) + 2*A057083(n) - 3*A057083(n-1)).

a(n) = (1/6)*(1 + 3^n + A099837(n+3) + A057682(n+3)). (End)