This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A094715 #27 Aug 28 2024 04:19:50
%S A094715 1,0,1,1,1,10,2,35,29,85,211,220,926,1001,3095,5461,9829,25126,37130,
%T A094715 97223,164921,349525,728575,1309528,2973350,5326685,11450531,22369621,
%U A094715 43942081,91869970,174174002,365088395,708653429,1431655765,2884834891
%N A094715 a(n) = Sum_{2*i+3*j=n, 0<=i<=n, 0<=j<=n} n!/( (2*i)!*(3*j)! ).
%C A094715 Average of binomial and inverse binomial transform of {1, 0, 0, 1, 0, 0, 1, ...}. - _Paul Barry_, Jan 04 2005
%H A094715 G. C. Greubel, <a href="/A094715/b094715.txt">Table of n, a(n) for n = 0..1000</a>
%F A094715 Limit_{n --> oo} a(n)/2^n = 1/6.
%F A094715 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
%F A094715 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]
%F A094715 E.g.f.: (exp(z) + 2*exp(-z/2)*cos(z*sqrt(3/4)))*cosh(z)/3. - _Peter Luschny_, Jul 10 2012
%F A094715 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
%p A094715 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
%t A094715 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 *)
%o A094715 (PARI) a(n)=sum(i=0,n,sum(j=0,n,if(n-2*i-3*j,0,n!/(2*i)!/(3*j)!)))
%o A094715 (Magma) R<x>:=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
%o A094715 (SageMath)
%o A094715 def A094715_list(prec):
%o A094715 P.<x> = PowerSeriesRing(ZZ, prec)
%o A094715 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()
%o A094715 A094715_list(50) # _G. C. Greubel_, Feb 13 2023
%Y A094715 Cf. A000748, A010892, A024493, A094717.
%K A094715 nonn
%O A094715 0,6
%A A094715 _Benoit Cloitre_, May 23 2004