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 A106188 #21 Sep 03 2021 12:26:19
%S A106188 1,2,7,22,77,274,1001,3706,13871,52326,198627,757758,2902783,11158358,
%T A106188 43019383,166275878,644099773,2499882098,9719235073,37845145898,
%U A106188 147565763893,576103020338,2251664727613,8809533747938,34499268410713
%N A106188 Expansion of 1/((1-x^2)*sqrt(1-4*x)).
%C A106188 Diagonal sums of number triangle A106187.
%H A106188 G. C. Greubel, <a href="/A106188/b106188.txt">Table of n, a(n) for n = 0..1000</a>
%F A106188 G.f. 1 / ((1 - x^2) * sqrt(1 - 4*x)).
%F A106188 a(n)=sum{k=0..floor(n/2), binomial(2(n-2k), n-2k)}.
%F A106188 PSUMSIGN transform of A006134. a(n+1) + a(n) = A006134(n). a(n) = Sum_{k=0..n} (-1)^k * binomial(2 * (n-k), n-k). - _Michael Somos_, Jun 20 2012
%F A106188 First difference is A054108. a(n+1) - a(n) = A054108(n). - _Michael Somos_, Jun 20 2012
%F A106188 D-finite with recurrence: n*a(n)+2*(1-2*n)*a(n-1) -n*a(n-2) +2*(2*n-1)*a(n-3)=0. - _R. J. Mathar_, Nov 09 2012
%F A106188 a(n) ~ 2^(2*n+4) / (15*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 03 2014
%e A106188 1 + 2*x + 7*x^2 + 22*x^3 + 77*x^4 + 274*x^5 + 1001*x^6 + 3706*x^7 + 13871*x^8 + ...
%t A106188 CoefficientList[Series[1/((1-x^2)*Sqrt[1-4*x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 03 2014 *)
%o A106188 (PARI) x='x+O('x^50); Vec(1/((1-x^2)*sqrt(1-4*x))) \\ _G. C. Greubel_, Mar 16 2017
%Y A106188 Cf. A006134, A054108. Convolution of A000984 and A059841.
%K A106188 easy,nonn
%O A106188 0,2
%A A106188 _Paul Barry_, Apr 24 2005