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 A108081 #39 Jan 19 2025 09:28:27
%S A108081 1,2,7,25,92,344,1300,4950,18955,72905,281403,1089343,4227273,
%T A108081 16438345,64037453,249855417,976205516,3818779616,14954876080,
%U A108081 58623077586,230007291334,903164858092,3549071519462,13955918890440,54912972103772,216194101316654,851622127750060
%N A108081 a(n) = Sum_{i=0..n} binomial(2*n-i, n+i).
%C A108081 A transform of the Fibonacci numbers A000045(n+1) under the mapping g(x)->(1/(c(x)*sqrt(1-4*x)))*g(x*c(x)), c(x) the g.f. of A000108. Hankel transform is the bisection of the Fibonacci numbers F(2*n+2) (A001906(n+1)). - _Paul Barry_, Sep 28 2007
%C A108081 Diagonal sums of A159965. - _Paul Barry_, Apr 28 2009
%C A108081 Comment from Li-yao Xia, Oct 22 2015: (Start)
%C A108081 Consider the smallest set X of finite sequences of integer (or words), such that
%C A108081 - 0 belongs to it;
%C A108081 - if a and b are two words in X, let L(a) be the word obtained by reversing a and subtracting 1 from every term, and R(b) be the word obtained by reversing b and adding 1 to every term; then the concatenations L(a).b and a.R(b) belong to X.
%C A108081 Examples of L and R values: L(10,30,20) = 19, 29, 9; R(10,30,20) = 21, 31, 11
%C A108081 List of words of X of lengths 1, 2, 3:
%C A108081 0
%C A108081 0, 1
%C A108081 -1, 0
%C A108081 -1, 0, 1 = L(0), 0, 1 = -1, 0, R(0)
%C A108081 0, 2, 1 = 0, R(0, 1)
%C A108081 1, -1, 0 = L(0), -1, 0
%C A108081 0, 1, 0 = 0, R(-1, 0)
%C A108081 0, -1, 0 = L(0, 1), 0
%C A108081 0, 1, 1 = 0, 1, R(0)
%C A108081 -1, -2, 0 = L(-1, 0), 0
%C A108081 The number of words of length n for n<=12 is given by a(n+1). Is this always true? (End)
%H A108081 G. C. Greubel and Vincenzo Librandi, <a href="/A108081/b108081.txt">Table of n, a(n) for n = 0..1000</a>(terms 0..200 from Vincenzo Librandi)
%F A108081 G.f.: 1/2*(1-5*x+4*x^2+((1-4*x)*(1-5*x)^2)^(1/2))/(1-4*x)/(1-4*x-x^2). - _Vladeta Jovovic_, Sep 06 2006
%F A108081 G.f.: (1+sqrt(1-4*x))/(2*sqrt(1-4*x)*(x+sqrt(1-4*x))). - _Paul Barry_, Sep 28 2007
%F A108081 a(n) = Sum_{k=0..n} C(n+k-1,k)*F(n-k+1). - _Paul Barry_, Sep 28 2007
%F A108081 Recurrence: n*(n+1)*a(n) = 2*(4*n^2 + 3*n - 6)*a(n-1) - (15*n^2 + 7*n - 48)*a(n-2) - 2*(n+2)*(2*n-3)*a(n-3). - _Vaclav Kotesovec_, Oct 17 2012
%F A108081 a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 17 2012
%F A108081 a(n) = [x^n] 1/((1-x-x^2) * (1-x)^n). - _Seiichi Manyama_, Apr 05 2024
%t A108081 CoefficientList[Series[(1+Sqrt[1-4*x])/(2*Sqrt[1-4*x]*(x+Sqrt[1-4*x])), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%t A108081 Table[Sum[Binomial[2n-i,n+i],{i,0,n}],{n,0,30}] (* _Harvey P. Dale_, Oct 20 2013 *)
%o A108081 (PARI) my(x='x+O('x^66)); Vec((1+sqrt(1-4*x))/(2*sqrt(1-4*x)*(x+sqrt(1-4*x)))) \\ _Joerg Arndt_, May 15 2013
%o A108081 (PARI) a(n) = sum(k=0,n, binomial(n+k-1,k)*fibonacci(n-k+1)); \\ _G. C. Greubel_, Jan 31 2017
%Y A108081 Cf. A000045, A000108, A001906, A159965.
%K A108081 nonn
%O A108081 0,2
%A A108081 _Ralf Stephan_, Jun 03 2005