A156266 a(n) = 7^n*Catalan(n).

1, 7, 98, 1715, 33614, 705894, 15529668, 353299947, 8243665430, 196199237234, 4744454282204, 116239129913998, 2879153833254412, 71978845831360300, 1813866914950279560, 46026872966863343835, 1175038992212864189670

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = 7^n*A000108(n).

From Gary W. Adamson, Jul 18 2011: (Start)

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

7, 7, 0, 0, 0, 0, ...

7, 7, 7, 0, 0, 0, ...

7, 7, 7, 7, 0, 0, ...

7, 7, 7, 7, 7, 0, ...

... (End)

E.g.f.: KummerM(1/2, 2, 28*x). - Peter Luschny, Aug 26 2012

G.f.: c(7*x) with c(x) the o.g.f. of A000108 (Catalan). - Philippe Deléham, Nov 15 2013

a(n) = Sum_{k=0..n} A085880(n,k)*6^k. - Philippe Deléham, Nov 15 2013

G.f.: 1/(1 - 7*x/(1 - 7*x/(1 - 7*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017

Sum_{n>=0} 1/a(n) = 266/243 + 392*arctan(1/(3*sqrt(3))) / (729*sqrt(3)). - Vaclav Kotesovec, Nov 23 2021

Sum_{n>=0} (-1)^n/a(n) = 770/841 - 1176*arctanh(1/sqrt(29)) / (841*sqrt(29)). - Amiram Eldar, Jan 25 2022

D-finite with recurrence (n+1)*a(n) +14*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

Maple

A156266_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
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

Mathematica

Table[7^n * CatalanNumber[n], {n, 0, 16}] (* Amiram Eldar, Jan 25 2022 *)

Prog

(Magma) [7^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

Crossrefs

Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128.

Column k=7 of A290605.

Sequence in context: A219406 A267664 A237022 * A234873 A051188 A367138

Adjacent sequences: A156263 A156264 A156265 * A156267 A156268 A156269

Keyword

nonn,easy

Author

Philippe Deléham, Feb 07 2009

Extensions

a(15) corrected by Vincenzo Librandi, Jul 19 2011

Status

approved