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 *)
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.
A256061
Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 2, 4, 0, 5, 30, 30, 0, 14, 196, 504, 336, 0, 42, 1260, 6300, 10080, 5040, 0, 132, 8184, 71280, 205920, 237600, 95040, 0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160, 0, 1430, 363220, 8288280, 58378320, 180180000, 273873600, 201801600, 57657600
Offset: 0
A(3,2) = 30: (())[], (()[]), (([])), ()()[], ()([]), ()[()], ()[[]], ()[](), ()[][], ([()]), ([[]]), ([]()), ([])(), ([])[], ([][]), [(())], [()()], [()[]], [()](), [()][], [([])], [[()]], [[]()], [[]](), [](()), []()(), []()[], []([]), [][()], [][]().
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 4;
0, 5, 30, 30;
0, 14, 196, 504, 336;
0, 42, 1260, 6300, 10080, 5040;
0, 132, 8184, 71280, 205920, 237600, 95040;
0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160;
...
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ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
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A[0, 0] = 1; A[n_, k_] := A[n, k] = k^n*CatalanNumber[n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 20 2017, translated from Maple *)
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
A291699
a(n) = n^n*(2*n)!/(n!*(n + 1)!).
Original entry on oeis.org
1, 1, 8, 135, 3584, 131250, 6158592, 353299947, 23991418880, 1883638417518, 167960000000000, 16772331868538246, 1854655886442627072, 225005916687384753700, 29718395534545380311040, 4245313393689422607421875, 652233889532678001886494720, 107247390031799133661006687830
Offset: 0
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seq(n^n*(2*n)!/n!/(n+1)!, n=0..50); # Robert Israel, Aug 30 2017
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Join[{1}, Table[n^n (2 n)!/(n! (n + 1)!), {n, 1, 17}]]
Table[SeriesCoefficient[2/(1 + Sqrt[1 - 4 n x]), {x, 0, n}], {n, 0, 17}]
Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]
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a(n)=binomial(2*n,n)/(n+1)*n^n \\ Charles R Greathouse IV, Oct 23 2023
A340968
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j*binomial(n,j)*Catalan(j).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 15, 1, 1, 5, 25, 71, 51, 1, 1, 6, 41, 199, 441, 188, 1, 1, 7, 61, 429, 1795, 2955, 731, 1, 1, 8, 85, 791, 5073, 17422, 20805, 2950, 1, 1, 9, 113, 1315, 11571, 64469, 177463, 151695, 12235, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
1, 5, 13, 25, 41, 61, ...
1, 15, 71, 199, 429, 791, ...
1, 51, 441, 1795, 5073, 11571, ...
1, 188, 2955, 17422, 64469, 181776, ...
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T_row := n -> k -> hypergeom([1/2, -n], [2], -4*k): for n from 0 to 6 do seq(simplify(T_row(n)(k)), k = 0..6) od; # Peter Luschny, Aug 27 2025
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T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * CatalanNumber[j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 01 2021 *)
A340968[n_, k_] := Hypergeometric2F1[1/2, -n, 2, -4*k]; Table[A340968[n, k], {n, 0, 6}, {k, 0, 7}] (* row-wise *) (* Peter Luschny, Aug 27 2025 *)
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T(n, k) = sum(j=0, n, k^j*binomial(n, j)*(2*j)!/(j!*(j+1)!));
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T(n, k) = 1+k*sum(j=0, n-1, T(j, k)*T(n-1-j, k));
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
Showing 1-8 of 8 results.
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