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 A108488 #25 Apr 23 2025 16:19:58
%S A108488 1,1,3,9,23,69,203,601,1815,5493,16731,51225,157367,485093,1499499,
%T A108488 4646233,14427095,44880981,139849979,436419737,1363713015,4266417221,
%U A108488 13362194571,41891406681,131452430999,412835452213,1297543367835
%N A108488 Expansion of 1/sqrt(1 -2*x -3*x^2 -4*x^3 +4*x^4).
%C A108488 In general, Sum_{k=0..n} C(n-k,k)^2*a^k*b^(n-k) has the expansion 1/sqrt(1 -2*b*x -(2*a*b -b^2)*x^2 -2*a*b^2*x^3 +(a*b)^2*x^4).
%C A108488 Diagonal of the rational function 1 / ((1 - x)*(1 - y) - 2*(x*y)^2). - _Ilya Gutkovskiy_, Apr 23 2025
%H A108488 G. C. Greubel, <a href="/A108488/b108488.txt">Table of n, a(n) for n = 0..1000</a>
%F A108488 a(n) = Sum_{k=0..n} C(n-k, k)^2*2^k.
%F A108488 a(n) ~ ((4*sqrt(2)-1)/62)^(1/4) * (1+2*sqrt(2)+sqrt(1+4*sqrt(2)))^(n+1) /(sqrt(Pi*n)*2^(n+2)). - _Vaclav Kotesovec_, Jul 24 2013
%F A108488 D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +3*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3) +4*(n-2)*a(n-4)=0. - _R. J. Mathar_, Aug 06 2013
%F A108488 G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * 2^k * x^k ). - _Paul D. Hanna_, Aug 31 2014
%t A108488 Table[Sum[Binomial[n-k,k]^2*2^k,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 24 2013 *)
%t A108488 CoefficientList[Series[1/Sqrt[1-2x-3x^2-4x^3+4x^4],{x,0,30}],x] (* _Harvey P. Dale_, Apr 06 2023 *)
%o A108488 (PARI) {a(n)=polcoeff( exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k) * 2^k * x^k) *x^m/m) +x*O(x^n)), n)}
%o A108488 for(n=0, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Aug 31 2014
%Y A108488 Cf. A051286, A108480.
%K A108488 easy,nonn
%O A108488 0,3
%A A108488 _Paul Barry_, Jun 04 2005