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%I #58 May 11 2024 16:36:36
%S 1,4,12,40,140,504,1848,6864,25740,97240,369512,1410864,5408312,
%T 20801200,80233200,310235040,1202160780,4667212440,18150270600,
%U 70690527600,275693057640,1076515748880,4208197927440,16466861455200,64495207366200,252821212875504,991837065896208
%N A Catalan transform of (1 + 2*x)/(1 - 2*x).
%C A Catalan transform of (1 + 2*x)/(1 - 2*x) under the mapping g(x) -> g(x*c(x)). (Here c(x) is the g.f. of A000108.) The original sequence can be retrieved by g(x) -> g(x*(1-x)).
%C Hankel transform is A144704. - _Paul Barry_, Sep 19 2008
%C Central terms of the triangle in A124927. - _Reinhard Zumkeller_, Mar 04 2012
%H Reinhard Zumkeller, <a href="/A100320/b100320.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%F G.f.: (1 + 2*x*c(x))/(1 - 2*x*c(x)), where c(x) is the g.f. of A000108.
%F a(n) = 4*binomial(2*n-1, n) - 3*0^n.
%F a(n) = binomial(2*n, n)*(4*2^(n-1) - 0^n)/2^n.
%F a(n) = Sum_{j=0..n} Sum_{k=0..n} C(2*n, n-k)*((2*k + 1)/(n + k + 1))*C(k, j)*(-1)^(j-k)*(4*2^(j-1) - 0^j).
%F a(n) = A028329(n), n > 0. - _R. J. Mathar_, Sep 02 2008
%F a(n) = T(2*n,n), where T(n,k) = A132046(n,k). - _Paul Barry_, Sep 19 2008
%F a(n) = Sum_{k=0..n} A039599(n,k)*A010684(k). - _Philippe Deléham_, Oct 29 2008
%F a(n) = A095660(2*n,n) for n > 0. - _Reinhard Zumkeller_, Apr 08 2012
%F G.f.: G(0) - 1, where G(k) = 1 + 1/(1 - 2*x*(2*k + 1)/(2*x*(2*k + 1) + (k + 1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013
%F a(n) = [x^n] (1 + 2*x)/(1 - x)^(n+1). - _Ilya Gutkovskiy_, Oct 12 2017
%F a(n) = 2*(2*n-1)*a(n-1)/n. - _G. C. Greubel_, Feb 01 2023
%F E.g.f.: 2*exp(2*x)*BesselI(0, 2*x) - 1. - _Stefano Spezia_, May 11 2024
%t a[0]= 1; a[n_]:= 2 Binomial[2 n, n];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 31 2018 *)
%o (Haskell)
%o a100320 n = a124927 (2 * n) n -- _Reinhard Zumkeller_, Mar 04 2012
%o (Magma) [4*Binomial(2*n-1, n) - 3*0^n: n in [0..40]]; // _G. C. Greubel_, Feb 01 2023
%o (SageMath)
%o def A100320(n): return 4*binomial(2*n-1, n) - 3*0^n
%o [A100320(n) for n in range(41)] # _G. C. Greubel_, Feb 01 2023
%Y Cf. A000108, A010684, A028329, A039599, A095660, A124927, A132046, A144704.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Nov 14 2004
%E Incorrect connection with A046055 deleted by _N. J. A. Sloane_, Jul 08 2009
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