A156270 - OEIS

%I #47 Sep 08 2022 08:45:41

%S 1,8,128,2560,57344,1376256,34603008,899678208,23991418880,

%T 652566593536,18034567675904,504967894925312,14294475794808832,

%U 408413594137395200,11762311511156981760,341107033823552471040,9952299339793060331520

%N a(n) = 8^n*Catalan(n).

%C A quarter of the count of And/Or-Trees with 2 variables [Chauvin]. - _R. J. Mathar_, Apr 01 2012

%H Vincenzo Librandi, <a href="/A156270/b156270.txt">Table of n, a(n) for n = 0..200</a>

%H Brigitte Chauvin, Philippe Flajolet, Daniele Gardy and Bernhard Gittenberger, <a href="http://dx.doi.org/10.1017/S0963548304006273">And/Or Tree Revisited</a>, Combinat., Probal. Comput., Vol. 13, No. 4-5 (2004), pp. 475-497.

%F a(n) = 8^n*A000108(n).

%F From _Gary W. Adamson_, Jul 18 2011: (Start)

%F a(n) is the upper left term in M^n, M = an infinite square production matrix:

%F   8, 8, 0, 0, 0, 0, ...

%F   8, 8, 8, 0, 0, 0, ...

%F   8, 8, 8, 8, 0, 0, ...

%F   8, 8, 8, 8, 8, 0, ...

%F   ... (End)

%F E.g.f.: KummerM(1/2, 2, 32*x). - _Peter Luschny_, Aug 26 2012

%F G.f.: c(8*x) with c(x) the o.g.f. of A000108 (Catalan). - _Philippe Deléham_, Nov 15 2013

%F a(n) = Sum_{k=0..n} A085880(n,k)*7^k. - _Philippe Deléham_, Nov 15 2013

%F G.f.: 1/(1 - 8*x/(1 - 8*x/(1 - 8*x/(1 - ...)))), a continued fraction. - _Ilya Gutkovskiy_, Aug 08 2017

%F (n+1)*a(n) +16*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Apr 14 2018

%F Sum_{n>=0} 1/a(n) = 1040/961 + 1536*arctan(1/sqrt(31)) / (961*sqrt(31)). - _Vaclav Kotesovec_, Nov 23 2021

%F Sum_{n>=0} (-1)^n/a(n) = 112/121 - 512*arctanh(1/sqrt(33)) / (363*sqrt(33)). - _Amiram Eldar_, Jan 25 2022

%F D-finite with recurrence +(n+1)*a(n) +16*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Mar 21 2022

%t Table[8^n*CatalanNumber[n], {n, 0, 20}] (* _Wesley Ivan Hurt_, Dec 28 2013 *)

%o (Magma) [8^n*Catalan(n): n in [0..20]]; // _Vincenzo Librandi_, Jul 19 2011

%Y Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128, A156266.

%Y Column k=8 of A290605.

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Feb 07 2009