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
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[6^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011
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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
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Table[CatalanNumber[n]6^n, {n, 0, 16}] (* Alonso del Arte, Jul 19 2011 *)
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
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[7^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011
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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
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Table[7^n * CatalanNumber[n], {n, 0, 16}] (* Amiram Eldar, Jan 25 2022 *)
A337167
a(n) = 1 + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).
Original entry on oeis.org
1, 4, 25, 199, 1795, 17422, 177463, 1870960, 20241403, 223438852, 2506596547, 28494103183, 327507800725, 3799735202218, 44440058006593, 523388751658831, 6201937444137619, 73888034816382820, 884517283667145259, 10634234680321209373, 128347834921058404249
Offset: 0
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a[n_] := a[n] = 1 + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 20}]
Table[Hypergeometric2F1[1/2, -n, 2, -12], {n, 0, 20}]
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{a(n) = sum(k=0, n, 3^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))} \\ Seiichi Manyama, Jan 31 2021
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my(N=20, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)*(1-13*x)))) \\ Seiichi Manyama, Feb 01 2021
A025226
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also a(n) = 3^n*C(n-1), where C = A000108 (Catalan numbers).
Original entry on oeis.org
3, 9, 54, 405, 3402, 30618, 288684, 2814669, 28146690, 287096238, 2975361012, 31241290626, 331638315876, 3553267670100, 38375290837080, 417331287853245, 4566095267100210
Offset: 1
a(3) = 3^3*C(2) = 27*2 = 54.
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[3^n*Catalan(n-1): n in [1..30]]; // G. C. Greubel, May 20 2022
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Rest[CoefficientList[Series[(1-Sqrt[1-12x])/2,{x,0,20}],x]] (* Harvey P. Dale, Mar 09 2011 *)
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a(n)=polcoeff((1-sqrt(1-12*x+x*O(x^n)))/2,n)
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[3^n*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, May 20 2022
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
- 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.
A098399
a(n) = 3^n*binomial(2*n+1, n).
Original entry on oeis.org
1, 9, 90, 945, 10206, 112266, 1250964, 14073345, 159497910, 1818276174, 20827527084, 239516561466, 2763652632300, 31979409030900, 370961144758440, 4312423307816865, 50227047938102310, 585982225944526950, 6846739692614999100, 80106854403595489470, 938394580156404305220
Offset: 0
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[3^n*Binomial(2*n+1, n): n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012
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Z:=(1-sqrt(1-3*z))*4^n/sqrt(1-3*z)/6: Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..18); # Zerinvary Lajos, Jan 01 2007
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Table[3^n Binomial[2n+1,n], {n,0,20}] (* Harvey P. Dale, Mar 28 2012 *)
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a(n)=binomial(2*n+1,n)*3^n \\ Charles R Greathouse IV, Oct 23 2023
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[3^n*binomial(2*n+1, n) for n in range(21)] # G. C. Greubel, Dec 27 2023
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
A337169
a(n) = (-1)^n + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).
Original entry on oeis.org
1, 2, 13, 89, 691, 5720, 49555, 443630, 4071595, 38105342, 362271823, 3488988101, 33967656469, 333752559392, 3305347855573, 32960499084305, 330664662067795, 3335002912108670, 33796042027030855, 343940115478559699, 3513702627928096681, 36021007341027948032
Offset: 0
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a[n_] := a[n] = (-1)^n + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 21}]
Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 12], {n, 0, 21}]
A156275
a(n) = 10^n*Catalan(n).
Original entry on oeis.org
1, 10, 200, 5000, 140000, 4200000, 132000000, 4290000000, 143000000000, 4862000000000, 167960000000000, 5878600000000000, 208012000000000000, 7429000000000000000, 267444000000000000000, 9694845000000000000000, 353576700000000000000000
Offset: 0
A217363
Series reversion of x - 3*x^3.
Original entry on oeis.org
1, 3, 27, 324, 4455, 66339, 1041012, 16953624, 283848543, 4855304025, 84482290035, 1490628232080, 26607713942628, 479621100042756, 8718235759397880, 159628084420459248, 2941328850997439439, 54501093415540789605, 1014898739548854163185
Offset: 1
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f:= k -> (3*k-3)!*3^(k-1)/(k-1)!/(2*k-1)!:
map(f, [$1..30]); # Robert Israel, May 07 2017
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CoefficientList[Series[2/3 Sqrt[z] Sin[ArcSin[(9 Sqrt[z])/2]/3], {z, 0, 20}], z](* Benedict W. J. Irwin, Jul 12 2016 *)
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