A156275 a(n) = 10^n*Catalan(n).

1, 10, 200, 5000, 140000, 4200000, 132000000, 4290000000, 143000000000, 4862000000000, 167960000000000, 5878600000000000, 208012000000000000, 7429000000000000000, 267444000000000000000, 9694845000000000000000, 353576700000000000000000

Offset

0,2

Comments

In general, for m >= 1, Sum_{k>=0} 1/(m^k * Catalan(k)) = 2*m*(8*m + 1) / (4*m - 1)^2 + 24 * m^2 * arcsin(1/(2*sqrt(m))) / (4*m - 1)^(5/2). - Vaclav Kotesovec, Nov 23 2021

Links

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

Vincent Pilaud, Pebble trees, arXiv:2205.06686 [math.CO], 2022.

Formula

a(n) = 10^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:

10, 10, 0, 0, 0, ...

10, 10, 10, 0, 0, ...

10, 10, 10, 10, 0, ...

10, 10, 10, 10, 10, ...

... (End)

E.g.f.: KummerM(1/2, 2, 40*x). - Peter Luschny, Aug 26 2012

G.f.: c(10*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)*9^k. - Philippe Deléham, Nov 15 2013

G.f.: 1/(1 - 10*x/(1 - 10*x/(1 - 10*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017

Sum_{n>=0} 1/a(n) = 180/169 + 800*arctan(1/sqrt(39)) / (507*sqrt(39)). - Vaclav Kotesovec, Nov 23 2021

Sum_{n>=0} (-1)^n/a(n) = 1580/1681 - 2400*arctanh(1/sqrt(41)) / (1681*sqrt(41)). - Amiram Eldar, Jan 25 2022

D-finite with recurrence (n+1)*a(n) +20*(-2*n+1)*a(n-1)=0. - ~~~

Mathematica

Table[10^n CatalanNumber[n], {n, 0, 20}] (* Harvey P. Dale, Mar 12 2013 *)

Prog

(Magma) [10^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

Crossrefs

Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128, A156266, A156270, A156273.

Column k=10 of A290605.

Sequence in context: A202436 A320671 A237025 * A036362 A051262 A367201

Adjacent sequences: A156272 A156273 A156274 * A156276 A156277 A156278

Keyword

nonn,easy

Author

Philippe Deléham, Feb 07 2009

Extensions

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

Status

approved