A094715 a(n) = Sum_{2*i+3*j=n, 0<=i<=n, 0<=j<=n} n!/( (2*i)!*(3*j)! ).

1, 0, 1, 1, 1, 10, 2, 35, 29, 85, 211, 220, 926, 1001, 3095, 5461, 9829, 25126, 37130, 97223, 164921, 349525, 728575, 1309528, 2973350, 5326685, 11450531, 22369621, 43942081, 91869970, 174174002, 365088395, 708653429, 1431655765, 2884834891

Offset: 0

Author

Benoit Cloitre, May 23 2004

Keywords

• nonn

Comments

Average of binomial and inverse binomial transform of {1, 0, 0, 1, 0, 0, 1, ...}. - Paul Barry, Jan 04 2005

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000748, A010892, A024493, A094717.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-2*x^2-x^3+x^4-x^5)/((1-2*x)*(1-x+x^2)*(1+3*x+3*x^2)) )); // G. C. Greubel, Feb 13 2023

Maple

A094715_list := proc(n) local i; (exp(z)+2*exp(-z/2)*cos(z*sqrt(3/4)))*cosh(z)/3;  series(%,z,n+2): seq(i!*coeff(%,z,i),i=0..n) end: A094715_list(34); # Peter Luschny, Jul 10 2012

Mathematica

Table[(1/6)*(Boole[n==0] +2^n +2*ChebyshevU[n,1/2] -ChebyshevU[n-1, 1/2] +2*3^(n/2)*ChebyshevU[n, -Sqrt[3]/2] +3^((n+1)/2)*ChebyshevU[n- 1, -Sqrt[3]/2]), {n,0,50}] (* G. C. Greubel, Feb 13 2023 *)

PARI

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

SageMath

def A094715_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x^2-x^3+x^4-x^5)/((1-2*x)*(1-x+x^2)*(1+3*x+3*x^2)) ).list()
A094715_list(50) # G. C. Greubel, Feb 13 2023

Formula

Limit_{n --> oo} a(n)/2^n = 1/6.

G.f.: (1-2*x^2-x^3+x^4-x^5)/((1-2*x)*(1-x+x^2)*(1+3*x+3*x^2)). - Vladeta Jovovic, May 23 2004

a(n) = (1/3)*Sum_{k=0..floor(n/2)} C(n, 2*k)*(2*cos(2*Pi*(n-2*k)/3) + 1). - Paul Barry, Jan 04 2005 [corrected by Jason Yuen, Aug 28 2024]

E.g.f.: (exp(z) + 2*exp(-z/2)*cos(z*sqrt(3/4)))*cosh(z)/3. - Peter Luschny, Jul 10 2012

a(n) = (1/6)*([n=0] + 2^n + 2*A010892(n) - A010892(n-1) + 2*A000748(n) + 3*A000748(n-1)). - G. C. Greubel, Feb 13 2023