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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.

A098405 Expansion of (1-sqrt(1-8*x))/((1-x)*(4*x*sqrt(1-8*x))).

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%I A098405 #23 Dec 31 2023 16:24:59
%S A098405 1,7,47,327,2343,17127,126951,950631,7173991,54471527,415652711,
%T A098405 3184708455,24485137255,188802730855,1459525454695,11307626564455,
%U A098405 87775235181415,682523302202215,5315297718995815,41450938169985895,323653082644384615,2529960757626047335
%N A098405 Expansion of (1-sqrt(1-8*x))/((1-x)*(4*x*sqrt(1-8*x))).
%C A098405 Partial sums of A069720.
%H A098405 Vincenzo Librandi, <a href="/A098405/b098405.txt">Table of n, a(n) for n = 0..200</a>
%F A098405 a(n) = Sum_{k=0..n} binomial(2*k+1, k)*2^k.
%F A098405 D-finite with recurrence: (n+1)*a(n) = (9*n+5)*a(n-1) - 4*(2*n+1)*a(n-2). - _Vaclav Kotesovec_, Oct 15 2012
%F A098405 a(n) ~ 2^(3*n+4)/(7*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 15 2012
%t A098405 Table[SeriesCoefficient[(1-Sqrt[1-8*x])/((1-x)*(4*x*Sqrt[1-8*x])),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 15 2012 *)
%t A098405 Accumulate[Table[2^(n-1) Binomial[2n-1,n],{n,20}]] (* _Harvey P. Dale_, Jan 20 2013 *)
%o A098405 (PARI) my(x='x+O('x^66)); Vec((1-sqrt(1-8*x))/((1-x)*(4*x*sqrt(1-8*x)))) \\ _Joerg Arndt_, May 11 2013
%o A098405 (Magma) [n le 2 select 7^(n-1) else ((9*n-4)*Self(n-1) - 4*(2*n-1)*Self(n-2))/n: n in [1..30]]; // _G. C. Greubel_, Dec 27 2023
%o A098405 (SageMath)
%o A098405 def A098405(n): return sum( binomial(2*k+1,k)*2^k for k in range(n+1))
%o A098405 [A098405(n) for n in range(41)] # _G. C. Greubel_, Dec 27 2023
%Y A098405 Cf. A069720.
%K A098405 easy,nonn
%O A098405 0,2
%A A098405 _Paul Barry_, Sep 06 2004