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%I #41 Sep 08 2022 08:45:41
%S 1,7,98,1715,33614,705894,15529668,353299947,8243665430,196199237234,
%T 4744454282204,116239129913998,2879153833254412,71978845831360300,
%U 1813866914950279560,46026872966863343835,1175038992212864189670
%N a(n) = 7^n*Catalan(n).
%H Vincenzo Librandi, <a href="/A156266/b156266.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = 7^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 7, 7, 0, 0, 0, 0, ...
%F 7, 7, 7, 0, 0, 0, ...
%F 7, 7, 7, 7, 0, 0, ...
%F 7, 7, 7, 7, 7, 0, ...
%F ... (End)
%F E.g.f.: KummerM(1/2, 2, 28*x). - _Peter Luschny_, Aug 26 2012
%F G.f.: c(7*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)*6^k. - _Philippe Deléham_, Nov 15 2013
%F G.f.: 1/(1 - 7*x/(1 - 7*x/(1 - 7*x/(1 - ...)))), a continued fraction. - _Ilya Gutkovskiy_, Aug 08 2017
%F Sum_{n>=0} 1/a(n) = 266/243 + 392*arctan(1/(3*sqrt(3))) / (729*sqrt(3)). - _Vaclav Kotesovec_, Nov 23 2021
%F Sum_{n>=0} (-1)^n/a(n) = 770/841 - 1176*arctanh(1/sqrt(29)) / (841*sqrt(29)). - _Amiram Eldar_, Jan 25 2022
%F D-finite with recurrence (n+1)*a(n) +14*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Mar 21 2022
%p A156266_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p for w from 1 to n do a[w] := 7*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od;convert(a,list)end: A156266_list(16); # _Peter Luschny_, May 19 2011
%t Table[7^n * CatalanNumber[n], {n, 0, 16}] (* _Amiram Eldar_, Jan 25 2022 *)
%o (Magma) [7^n*Catalan(n): n in [0..20]]; // _Vincenzo Librandi_, Jul 19 2011
%Y Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128.
%Y Column k=7 of A290605.
%K nonn,easy
%O 0,2
%A _Philippe Deléham_, Feb 07 2009
%E a(15) corrected by _Vincenzo Librandi_, Jul 19 2011
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