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A064613 Second binomial transform of the Catalan numbers.

%I A064613 #90 Jan 20 2025 10:51:38
%S A064613 1,3,10,37,150,654,3012,14445,71398,361114,1859628,9716194,51373180,
%T A064613 274352316,1477635912,8016865533,43773564294,240356635170,
%U A064613 1326359740956,7351846397334,40913414754324,228508350629892
%N A064613 Second binomial transform of the Catalan numbers.
%C A064613 Exponential convolution of Catalan numbers and powers of 2. - _Vladeta Jovovic_, Dec 03 2004
%C A064613 Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...]. - _Philippe Deléham_, Oct 24 2007
%C A064613 a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and those at a higher level come in 4 colors. Example: a(3)=37 because, denoting  U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, 3 paths of shape UDH, and 4 paths of shape UHD. - _Emeric Deutsch_, May 02 2011
%C A064613 a(n) is the number of Schroeder paths of semilength n in which the (2,0)-steps come in 2 colors and having no (2,0)-steps at levels 1,3,5,... - _José Luis Ramírez Ramírez_, Mar 30 2013
%C A064613 From _Tom Copeland_, Nov 08 2014: (Start)
%C A064613 This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t)=x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2; and its inverse Cinv(x) = x*(1-x). (Cf A126930.)
%C A064613 O.g.f.: G(x) = C[P[P(x,-1),-1]] = C[P(x,-2)] = (1-sqrt(1-4*x/(1-2*x)))/2 = x*A064613(x).
%C A064613 Ginv(x) =  Pinv[Cinv(x),-2] = P[Cinv(x),2] = x(1-x)/[1+2x(1-x)] = (x-x^2)/[1+2(x-x^2)] = x - 3 x^2 + 8 x^3 - ... is -A155020(-x) ignoring first term there. (Cf. A146559, A125145.)(End)
%H A064613 G. C. Greubel, <a href="/A064613/b064613.txt">Table of n, a(n) for n = 0..1000</a>
%H A064613 Isaac DeJager, Madeleine Naquin, and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H A064613 Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv:1110.6638 [math.NT], 2011.
%H A064613 Paveł Szabłowski, <a href="https://cdm.ucalgary.ca/article/view/76214">Beta distributions whose moment sequences are related to integer sequences listed in the OEIS</a>, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 98.
%F A064613 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k, k)*2^(n-k)/(k+1).
%F A064613 a(n) = 2^n*hypergeom([1/2, -n], [2], -2).
%F A064613 G.f.: (1-sqrt((1-6*x)/(1-2*x)))/(2*x). - _Vladeta Jovovic_, May 03 2003
%F A064613 With offset 1: a(1) = 1, a(n) = 2^(n-1) + Sum_{i=1..n-1} a(i)*a(n-i). - _Benoit Cloitre_, Mar 16 2004
%F A064613 D-finite with recurrence (n+1)*a(n) = (8*n-2)*a(n-1) - (12*n-12)*a(n-2). - _Vladeta Jovovic_, Jul 16 2004
%F A064613 E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x)). - _Vladeta Jovovic_, Dec 03 2004
%F A064613 Inverse binomial transform of A104455. - _Philippe Deléham_, Nov 30 2007
%F A064613 G.f.: 1/(1-3*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-... (continued fraction). - _Paul Barry_, Jul 02 2009
%F A064613 a(n) = Sum_{0<=k<=n} A052179(n,k)*(-1)^k. - _Philippe Deléham_, Nov 28 2009
%F A064613 From _Gary W. Adamson_, Jul 21 2011: (Start)
%F A064613 a(n) = the upper left term in M^n, M = an infinite square production matrix as follows:
%F A064613   3, 1, 0, 0, ...
%F A064613   1, 3, 1, 0, ...
%F A064613   1, 1, 3, 1, ...
%F A064613   1, 1, 1, 3, ...
%F A064613   ... (End)
%F A064613 a(n) ~ 2^(n-3/2)*3^(n+3/2)/(n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Jun 29 2013
%F A064613 G.f. A(x) satisfies: A(x) = 1/(1 - 2*x) + x * A(x)^2. - _Ilya Gutkovskiy_, Jun 30 2020
%t A064613 CoefficientList[Series[(1-Sqrt[(1-6*x)/(1-2*x)])/2/x, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 29 2013 *)
%t A064613 a[n_] := 2^n Hypergeometric2F1[1/2, -n, 2, -2];
%t A064613 Array[a, 22, 0] (* _Peter Luschny_, Jan 27 2020 *)
%o A064613 (PARI) x='x+O('x^66); Vec((1-sqrt((1-6*x)/(1-2*x)))/(2*x)) /* _Joerg Arndt_, Mar 31 2013 */
%o A064613 (Magma) I:=[3,10]; [1] cat [n le 2 select I[n] else ((8*n-2)*Self(n-1)-(12*n-12)*Self(n-2))div (n+1): n in [1..30]]; // _Vincenzo Librandi_, Jan 23 2017
%Y A064613 Cf. A007317, A000108, A014318.
%Y A064613 Cf. A125145, A146559, A126930.
%K A064613 nonn
%O A064613 0,2
%A A064613 _Karol A. Penson_, Sep 24 2001
%E A064613 Name clarified using a comment of _Vladeta Jovovic_ by _Peter Bala_, Jan 27 2020