A085880 -id:A085880 - cp's OEIS frontend

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

Philippe Deléham, Feb 07 2009

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vincent Pilaud, Pebble trees, arXiv:2205.06686 [math.CO], 2022.

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

Column k=10 of A290605.

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

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

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. - R. J. Mathar, Mar 21 2022

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

A360651 Triangle T(n, m) = (n - m + 1)C(2n + 1, m)C(2n - m + 2, n - m + 1)/(2*n - m + 2).

Original entry on oeis.org

1, 3, 3, 10, 20, 10, 35, 105, 105, 35, 126, 504, 756, 504, 126, 462, 2310, 4620, 4620, 2310, 462, 1716, 10296, 25740, 34320, 25740, 10296, 1716, 6435, 45045, 135135, 225225, 225225, 135135, 45045, 6435, 24310, 194480, 680680, 1361360, 1701700, 1361360, 680680, 194480, 24310

Offset: 0

Vladimir Kruchinin, Feb 15 2023

			Triangle T(n, m) starts:
[0] 1;
[1] 3,     3;
[2] 10,    20,    10;
[3] 35,    105,   105,     35;
[4] 126,   504,   756,     504,     126;
[5] 462,   2310,  4620,    4620,    2310,    462;
[6] 1716,  10296, 25740,   34320,   25740,   10296,    1716;
[7] 6435,  45045, 135135,  225225,  225225,  135135,   45045,  6435;

Cf. A001700, A085880, A069720 (row sums).

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
T := (n, k) -> (2*n + 1)*CatalanNumber(n)*binomial(n, k):
seq(seq(T(n, k), k = 0..n), n = 0..8); # Peter Luschny, Feb 15 2023

Maxima
```
T(n,m):=if n
				
```

G.f.: 2/(1 - 4*x + sqrt(1 - 4*x - 4*x*y) - 4*x*y).

T(n, k) = binomial(n, k)*CatalanNumber(n)*(2*n + 1). - Peter Luschny, Feb 15 2023

cp's OEIS Frontend

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

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A360651 Triangle T(n, m) = (n - m + 1)C(2n + 1, m)C(2n - m + 2, n - m + 1)/(2*n - m + 2).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A360651 Triangle T(n, m) = (n - m + 1)*C(2*n + 1, m)*C(2*n - m + 2, n - m + 1)/(2*n - m + 2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Maxima

Formula

A360651 Triangle T(n, m) = (n - m + 1)C(2n + 1, m)C(2n - m + 2, n - m + 1)/(2*n - m + 2).