A151403 -id:A151403 - cp's OEIS frontend

A085880 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

1, 1, 1, 2, 4, 2, 5, 15, 15, 5, 14, 56, 84, 56, 14, 42, 210, 420, 420, 210, 42, 132, 792, 1980, 2640, 1980, 792, 132, 429, 3003, 9009, 15015, 15015, 9009, 3003, 429, 1430, 11440, 40040, 80080, 100100, 80080, 40040, 11440, 1430, 4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862

Offset: 0

N. J. A. Sloane, Aug 17 2003

Coefficients of terms in the series reversion of (1-k*x-(k+1)*x^2)/(1+x). - Paul Barry, May 21 2005
Equals A131427 * A007318 as infinite lower triangular matrices. [Philippe Deléham, Sep 15 2008]
Sum_{k=0..n} T(n,k)*x^k = A168491(n), A000007(n), A000108(n), A151374(n), A005159(n), A151403(n), A156058(n), A156128(n), A156266(n), A156270(n), A156273(n), A156275(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Nov 15 2013
Diagonal sums are A052709(n+1). - Philippe Deléham, Nov 15 2013

			Triangle starts:
[ 1]     1;
[ 2]     1,     1;
[ 3]     2,     4,      2;
[ 4]     5,    15,     15,      5;
[ 5]    14,    56,     84,     56,     14;
[ 6]    42,   210,    420,    420,    210,     42;
[ 7]   132,   792,   1980,   2640,   1980,    792,    132;
[ 8]   429,  3003,   9009,  15015,  15015,   9009,   3003,    429;
[ 9]  1430, 11440,  40040,  80080, 100100,  80080,  40040,  11440,  1430;
[10]  4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.

Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Binomial(2*n,n)/( n+1) ))); # G. C. Greubel, Feb 07 2020

[Binomial(n,k)*Catalan(n): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 07 2020

seq(seq(binomial(n, k)*binomial(2*n, n)/(n+1), k = 0..n), n = 0..10); # G. C. Greubel, Feb 07 2020

Table[Binomial[n, k]*CatalanNumber[n], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2020 *)

tabl(nn) = {for (n=0, nn, c =  binomial(2*n,n)/(n+1); for (k=0, n, print1(c*binomial(n, k), ", ");); print(););} \\ Michel Marcus, Apr 09 2015

[[binomial(n,k)*catalan_number(n) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 07 2020

Triangle given by [1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
Sum_{k>=0} T(n, k) = A151374(n) (row sums). - Philippe Deléham, Aug 11 2005
G.f.: (1-sqrt(1-4*(x+y)))/(2*(x+y)). - Vladimir Kruchinin, Apr 09 2015

A156058 a(n) = 5^n * Catalan(n).

Original entry on oeis.org

1, 5, 50, 625, 8750, 131250, 2062500, 33515625, 558593750, 9496093750, 164023437500, 2870410156250, 50784179687500, 906860351562500, 16323486328125000, 295863189697265625, 5395152282714843750, 98911125183105468750, 1822047042846679687500

Offset: 0

Philippe Deléham, Feb 03 2009

From Joerg Arndt, Oct 22 2012: (Start)
Number of strings of length 2*n of five different types of balanced parentheses.
The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)
Number of Dyck paths of length 2n in which the step U=(1,1) come in 5 colors. [José Luis Ramírez Ramírez, Jan 31 2013]

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

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

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

A156058_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 5*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od; convert(a,list)end: A156058_list(16); # Peter Luschny, May 19 2011
A156058 := proc(n)
    5^n*A000108(n) ;
end proc: # R. J. Mathar, Oct 06 2012

Table[5^n CatalanNumber[n],{n,0,20}]  (* Harvey P. Dale, Mar 13 2011 *)

a(n) = 5^n*A000108(n).
From Gary W. Adamson, Jul 18 2011: (Start)
a(n) is the upper left term in M^n, M = the infinite square production matrix:
  5, 5, 0, 0, 0, 0,...
  5, 5, 5, 0, 0, 0,...
  5, 5, 5, 5, 0, 0,...
  5, 5, 5, 5, 5, 0,...
  ... (End)
E.g.f.: KummerM(1/2, 2, 20*x). - Peter Luschny, Aug 26 2012
D-finite with recurrence (n+1)*a(n) -10*(2*n-1)*a(n-1)=0. - R. J. Mathar, Oct 06 2012
G.f.: c(5*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)*4^k. - Philippe Deléham, Nov 15 2013
G.f.: 1/(1 - 5*x/(1 - 5*x/(1 - 5*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Apr 19 2017
Sum_{n>=0} 1/a(n) = 410/361 + 600*arctan(1/sqrt(19)) / (361*sqrt(19)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 130/147 - 200*arctanh(1/sqrt(21)) / (147*sqrt(21)). - Amiram Eldar, Jan 25 2022

A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4kx)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 40, 14, 0, 1, 5, 32, 135, 224, 42, 0, 1, 6, 50, 320, 1134, 1344, 132, 0, 1, 7, 72, 625, 3584, 10206, 8448, 429, 0, 1, 8, 98, 1080, 8750, 43008, 96228, 54912, 1430, 0, 1, 9, 128, 1715, 18144, 131250, 540672, 938223, 366080, 4862, 0

Offset: 0

Ilya Gutkovskiy, Aug 07 2017

Number of 2n-length strings of balanced parentheses of at most k different types. Also number of binary trees with n inner nodes of at most k different dimensions. - Alois P. Heinz, Oct 28 2019

			G.f. of column k: A(x) = 1 + k*x + 2*k^2*x^2 + 5*k^3*x^3 + 14*k^4*x^4 + 42*k^5*x^5 + 132*k^6*x^6 + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   2,     8,     18,     32,      50,  ...
  0,   5,    40,    135,    320,     625,  ...
  0,  14,   224,   1134,   3584,    8750,  ...
  0,  42,  1344,  10206,  43008,  131250,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give:  A000007, A000108, A151374, A005159, A151403, A156058, A156128, A156266, A156270, A156273, A156275.
Rows n=0-2 give: A000012, A001477, A001105.
Main diagonal gives A291699.
Cf. A000108, A253180, A256061.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Oct 28 2019

Table[Function[k, SeriesCoefficient[2/(1 + Sqrt[1 - 4 k x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

A(n,k) = k^n*(2*n)!/(n!*(n + 1)!).
A(n,k) = k^n*A000108(n).
G.f. of column k: 2/(1 + sqrt(1 - 4*k*x)).
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - ...)))))), a continued fraction.
E.g.f. of column k: (BesselI(0,2*k*x) - BesselI(1,2*k*x))*exp(2*k*x).
If g.f. = 2/(1 + sqrt(1 - 4*k*x)), then a(n) ~ k^n*4^n/(sqrt(Pi)*n^(3/2)).
A(n,k) = Sum_{i=0..k} binomial(k,i) * A256061(n,k-i). - Alois P. Heinz, Oct 28 2019
For fixed k >= 1, Sum_{n>=0} 1/A(n,k) = 2*k*(8*k + 1) / (4*k - 1)^2 + 24 * k^2 * arcsin(1/(2*sqrt(k))) / (4*k - 1)^(5/2). - Vaclav Kotesovec, Nov 23 2021
For fixed k >= 1, Sum_{n>=0} (-1)^n / A(n,k) = 2*k*(8*k - 1) / (4*k + 1)^2 - 24 * k^2 * log((1 + sqrt(4*k + 1))/(2*sqrt(k))) / (4*k + 1)^(5/2). - Vaclav Kotesovec, Nov 24 2021

A156128 a(n) = 6^n * Catalan(n).

Original entry on oeis.org

1, 6, 72, 1080, 18144, 326592, 6158592, 120092544, 2401850880, 48997757952, 1015589892096, 21327387734016, 452796847276032, 9702789584486400, 209580255024906240, 4558370546791710720, 99747873141559787520, 2194453209114315325440, 48508965675158549299200

Offset: 0

Philippe Deléham, Feb 04 2009

Number of Dyck n-paths with two types of up step and three types of down step. - David Scambler, Jun 21 2013

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

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

Column k=6 of A290605.

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

A156128_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 6*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od;convert(a,list)end: A156128_list(16); # Peter Luschny, May 19 2011

Table[CatalanNumber[n]6^n, {n, 0, 16}] (* Alonso del Arte, Jul 19 2011 *)

a(n) = 6^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:
  6, 6, 0, 0, 0, 0, ...
  6, 6, 6, 0, 0, 0, ...
  6, 6, 6, 6, 0, 0, ...
  6, 6, 6, 6, 6, 0, ...
  ... (End)
E.g.f.: KummerM(1/2, 2, 24*x). - Peter Luschny, Aug 26 2012
G.f.: c(6*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) * 5^k. - Philippe Deléham, Nov 15 2013
G.f.: 1/(1 - 6*x/(1 - 6*x/(1 - 6*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017
Sum_{n>=0} 1/a(n) = 588/529 + 864*arctan(1/sqrt(23)) / (529*sqrt(23)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 564/625 - 432*log(3/2) / 3125. - Amiram Eldar, Jan 25 2022
D-finite with recurrence (n+1)*a(n) +12*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

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

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Feb 07 2009

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

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

Column k=7 of A290605.

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

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

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

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

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

A151254 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.

Original entry on oeis.org

1, 4, 20, 96, 480, 2368, 11840, 58880, 294400, 1468416, 7342080, 36667392, 183336960, 916144128, 4580720640, 22896574464, 114482872320, 572320645120, 2861603225600, 14306741583872, 71533707919360, 357650927714304, 1788254638571520, 8941026626502656, 44705133132513280, 223522175800311808

Offset: 0

Manuel Kauers, Nov 18 2008

Hankel transform is 4^binomial(n+1,2). - Philippe Deléham, Feb 01 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4
Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2009.

Cf. A000108, A001405, A120730, A151162, A151281, A151403, A156058.

[n le 3 select Factorial(n+2)/6 else (5*n*Self(n-1) + 16*(n-3)*Self(n-2) - 80*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

aux[i_, j_, k_, n_]:= Which[Min[i, j, k, n]<0 || Max[i, j, k]>n, 0, n==0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1+i, -1+j, -1+k, -1+n] + aux[-1+i, -1+j, k, -1+n] + aux[-1+i, j, -1+k, -1+n] + aux[-1+i, j, k, -1 + n] + aux[1+i, j, k, -1+n]]; Table[Sum[aux[i,j,k,n], {i,0,n}, {j,0,n}, {k,0,n}], {n, 0, 30}]
a[n_]:= a[n]= If[n<3, (n+3)!/3!, (5*(n+1)*a[n-1] +16*(n-2)*a[n-2] -80*(n-2)*a[n- 3])/(n+1)]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 09 2022 *)

def a(n): # a = A151254
    if (n==0): return 1
    elif (n%2==1): return 5*a(n-1) - 4^((n-1)/2)*catalan_number((n-1)/2)
    else: return 5*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum_{k=0..n} A120730(n,k)*4^k. - Philippe Deléham, Feb 01 2009
From Philippe Deléham, Feb 02 2009: (Start)
a(2n+2) = 5*a(2n+1), a(2n+1) = 5*a(2n) - 4^n*A000108(n) = 5*a(2n) - A151403(n).
G.f.: (sqrt(1-16*x^2) + 8*x - 1)/(8*x*(1-5*x)). (End)
a(n) = (5*(n+1)*a(n-1) + 16*(n-2)*a(n-2) - 80*(n-2)*a(n-3))/(n+1). - G. C. Greubel, Nov 09 2022

A303537 Expansion of ((1 + 4x)/(1 - 4x))^(1/4).

Original entry on oeis.org

1, 2, 2, 12, 22, 124, 276, 1496, 3686, 19436, 51068, 263720, 724860, 3681880, 10466920, 52450992, 153093254, 758495564, 2261603564, 11096526344, 33676743956, 163842737928, 504738342808, 2437418983888, 7605947276508, 36487283224952, 115140704639576

Offset: 0

Seiichi Manyama, Apr 25 2018

nonn

Let ((1 + k*x)/(1 - k*x))^(1/k) = a(0) + a(1)*x + a(2)*x^2 + ...

Then n*a(n) = 2*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A063886, A081654, A151403, A303538.

CoefficientList[Series[Surd[(1+4x)/(1-4x),4],{x,0,40}],x] (* Harvey P. Dale, Jul 25 2021 *)

N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(1/4))

a(n) ~ 2^(2*n + 1/4) / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 26 2018
n*a(n) = 2*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.
G.f.: A(x)=F(x*G(x^2)), where F(x) is the g.f. for A063886, and G(x) is the g.f. for A151403. - Alexander Burstein, Nov 13 2023

A098400 a(n) = 4^nbinomial(2n+1, n).

Original entry on oeis.org

1, 12, 160, 2240, 32256, 473088, 7028736, 105431040, 1593180160, 24216338432, 369849532416, 5671026163712, 87246556364800, 1346089726771200, 20819521107394560, 322702577164615680, 5011381198321090560, 77954818640550297600, 1214454016715941478400

Offset: 0

Paul Barry, Sep 06 2004

G. C. Greubel, Table of n, a(n) for n = 0..820

Cf. A000108, A001700, A069720, A069723, A098399, A151403.

[4^n*(2*n+1)*Catalan(n): n in [0..30]]; // G. C. Greubel, Dec 27 2023

Table[4^n Binomial[2n+1,n],{n,0,20}] (* Harvey P. Dale, Jan 22 2019 *)

a(n)=binomial(2*n+1,n)<<(2*n) \\ Charles R Greathouse IV, Oct 23 2023

[4^n*binomial(2*n+1,n) for n in range(31)] # G. C. Greubel, Dec 27 2023

G.f.: (1-sqrt(1-16*x))/(8*x*sqrt(1-16*x)).
E.g.f.: a(n) = n! * [x^n] exp(8*x)*(BesselI(0, 8*x) + BesselI(1, 8*x)). - Peter Luschny, Aug 25 2012
(n+1)*a(n) - 8*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 26 2012
a(n) = 4^n*(2*n+1)*Hypergeometric2F1([1-n,-n],[2],1). - Peter Luschny, Sep 22 2014
From G. C. Greubel, Dec 27 2023: (Start)
a(n) = 4^n * A001700(n).
a(n) = 4^n * (2*n+1) * A000108(n).
a(n) = (2*n+1) * A151403(n). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 8/15 + 128*arcsin(1/4)/(15*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 8/17 + 128*arcsinh(1/4)/(17*sqrt(17)). (End)

A156270 a(n) = 8^n*Catalan(n).

Original entry on oeis.org

1, 8, 128, 2560, 57344, 1376256, 34603008, 899678208, 23991418880, 652566593536, 18034567675904, 504967894925312, 14294475794808832, 408413594137395200, 11762311511156981760, 341107033823552471040, 9952299339793060331520

Offset: 0

Philippe Deléham, Feb 07 2009

A quarter of the count of And/Or-Trees with 2 variables [Chauvin]. - R. J. Mathar, Apr 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Brigitte Chauvin, Philippe Flajolet, Daniele Gardy and Bernhard Gittenberger, And/Or Tree Revisited, Combinat., Probal. Comput., Vol. 13, No. 4-5 (2004), pp. 475-497.

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

Column k=8 of A290605.

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

Table[8^n*CatalanNumber[n], {n, 0, 20}] (* Wesley Ivan Hurt, Dec 28 2013 *)

a(n) = 8^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:
  8, 8, 0, 0, 0, 0, ...
  8, 8, 8, 0, 0, 0, ...
  8, 8, 8, 8, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ... (End)
E.g.f.: KummerM(1/2, 2, 32*x). - Peter Luschny, Aug 26 2012
G.f.: c(8*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)*7^k. - Philippe Deléham, Nov 15 2013
G.f.: 1/(1 - 8*x/(1 - 8*x/(1 - 8*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017
(n+1)*a(n) +16*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Apr 14 2018
Sum_{n>=0} 1/a(n) = 1040/961 + 1536*arctan(1/sqrt(31)) / (961*sqrt(31)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 112/121 - 512*arctanh(1/sqrt(33)) / (363*sqrt(33)). - Amiram Eldar, Jan 25 2022
D-finite with recurrence +(n+1)*a(n) +16*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

A156273 a(n) = 9^n*Catalan(n).

Original entry on oeis.org

1, 9, 162, 3645, 91854, 2480058, 70150212, 2051893701, 61556811030, 1883638417518, 58564030799196, 1844766970174674, 58748732742485772, 1888352123865614100, 61182608813245896840, 1996082612532147384405, 65518476340761072970470, 2162109719245115408025510

Offset: 0

Philippe Deléham, Feb 07 2009

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

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

Column k=9 of A290605.

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

Table[9^n CatalanNumber[n],{n,0,20}] (* Harvey P. Dale, Sep 09 2012 *)

a(n) = 9^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:
  9, 9, 0, 0, 0, 0, ...
  9, 9, 9, 0, 0, 0, ...
  9, 9, 9, 9, 0, 0, ...
  9, 9, 9, 9, 9, 0, ...
  ... (End)
E.g.f.: KummerM(1/2, 2, 36*x). - Peter Luschny, Aug 26 2012
G.f.: c(9*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)*8^k. - Philippe Deléham, Nov 15 2013
G.f.: 1/(1 - 9*x/(1 - 9*x/(1 - 9*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017
Sum_{n>=0} 1/a(n) = 1314/1225 + 1944*arctan(1/sqrt(35)) / (1225*sqrt(35)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 1278/1369 - 1944*arctanh(1/sqrt(37)) / (1369*sqrt(37)). - Amiram Eldar, Jan 25 2022
D-finite with recurrence (n+1)*a(n) +18*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

cp's OEIS Frontend

A085880 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

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A156058 a(n) = 5^n * Catalan(n).

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A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4*k*x)).

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A156128 a(n) = 6^n * Catalan(n).

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

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A151254 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.

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A303537 Expansion of ((1 + 4*x)/(1 - 4*x))^(1/4).

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A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4kx)).

A303537 Expansion of ((1 + 4x)/(1 - 4x))^(1/4).

A098400 a(n) = 4^nbinomial(2n+1, n).