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.
A369102
Expansion of (1/x) * Series_Reversion( x * ((1-x)^4-x^4) ).
Original entry on oeis.org
1, 4, 26, 204, 1772, 16408, 158752, 1585968, 16235472, 169423232, 1795611168, 19275231872, 209140483328, 2289981517312, 25271472702464, 280795784911616, 3138701648319744, 35270318924758016, 398215386792574464, 4515067063939210240, 51388662166213954560
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^4-x^4))/x)
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a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(5*n+3, n-4*k))/(n+1);
A138020
G.f. satisfies A(x) = sqrt( (1 + 2*x*A(x)) / (1 - 2*x*A(x)) ).
Original entry on oeis.org
1, 2, 6, 24, 110, 544, 2828, 15232, 84246, 475648, 2730068, 15882240, 93438540, 554967040, 3323125528, 20039827456, 121597985254, 741871845376, 4548193111428, 28004975116288, 173113004348580, 1073893324357632, 6683288759506856, 41715337804120064
Offset: 0
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A138020 := proc(n)
option remember ;
if n < 5 then
op(n+1,[1,2,6,24,110]) ;
else
4*(-55*n^3 +231*n^2 -263*n +51)*procname(n-2) -16*(n-3)*(n-4)*(5*n-1)*procname(n-4) ;
-%/n/(n+1)/(5*n-11)
end if;
end proc:
seq(A138020(n),n=0..30) ; # R. J. Mathar, Sep 27 2024
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CoefficientList[y/.AsymptoticSolve[y^2-1-2x(y+y^3) ==0,y->1,{x,0,23}][[1]],x] (* Alexander Burstein, Nov 26 2021 *)
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a(n)=polcoeff((1/x)*serreverse(x*sqrt((1-2*x)/(1+2*x+x^2*O(x^n)))),n)
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
A354733
a(0) = a(1) = 1; a(n) = 2 * Sum_{k=0..n-2} a(k) * a(n-k-2).
Original entry on oeis.org
1, 1, 2, 4, 10, 24, 64, 168, 464, 1280, 3624, 10304, 29728, 86240, 252480, 743040, 2200640, 6547200, 19571200, 58727680, 176883200, 534476800, 1619912320, 4923070464, 14999764480, 45807916544, 140196076544, 429931051008, 1320905583616, 4065358827520
Offset: 0
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a[0] = a[1] = 1; a[n_] := a[n] = 2 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 29}]
nmax = 29; CoefficientList[Series[(1 - Sqrt[1 - 8 x^2 (1 + x)])/(4 x^2), {x, 0, nmax}], x]
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a(n) = sum(k=0, n\2, 2^k*binomial(k+1, n-2*k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Nov 05 2023
A200375
Product of Catalan and Jacobsthal numbers: a(n) = A000108(n)*A001045(n+1).
Original entry on oeis.org
1, 1, 6, 25, 154, 882, 5676, 36465, 244530, 1657942, 11471668, 80242890, 568080772, 4056976900, 29212908120, 211783889025, 1544811959970, 11328491394990, 83473572128100, 617702666484750, 4588654943721420, 34206312386929020, 255803818897858920, 1918528298674328250, 14427334095935095764
Offset: 0
G.f.: A(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 + 429*85*x^7 + 1430*171*x^8 +...+ A000108(n)*A001045(n)*x^n +...
The g.f. of the Jacobsthal sequence A001045, F(x) = 1/(1-x-2*x^2), begins:
F(x) = 1 + x + 3*x^2 + 5*x^3 + 11*x^4 + 21*x^5 + 43*x^6 + 85*x^7 + 171*x^8 +...
The g.f. of A200376, where G(x) = A(x/G(x)), begins:
G(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...
in which the odd-indexed coefficients are powers of 9.
- Michael De Vlieger, Table of n, a(n) for n = 0..1112
- Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
- Paul Barry and Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.
- S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
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Array[CatalanNumber[# - 1] (2^# - (-1)^#)/3 &, 25] (* Michael De Vlieger, Apr 24 2018 *)
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{a(n) = binomial(2*n, n)/(n+1) * (2^(n+1) + (-1)^n)/3}
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{a(n) = polcoef(sqrt((1-2*x - sqrt(1-4*x-32*x^2 +O(x^(n+3))))/2)/(3*x), n)}
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{a(n) = polcoef((1/x)*serreverse(x-x^2 - 4*x^3*sum(m=0,n\2,binomial(2*m,m)/(m+1)*3^m*x^(2*m)) +x^3*O(x^n)), n)}
A337168
a(n) = (-1)^n + 2 * Sum_{k=0..n-1} a(k) * a(n-k-1).
Original entry on oeis.org
1, 1, 5, 21, 105, 553, 3053, 17405, 101713, 606033, 3667797, 22485477, 139340985, 871429497, 5492959293, 34862161869, 222592918689, 1428814897825, 9215016141989, 59684122637237, 388045493943049, 2531696701375689, 16569559364596365, 108758426952823709
Offset: 0
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a[n_] := a[n] = (-1)^n + 2 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k CatalanNumber[k], {k, 0, n}], {n, 0, 23}]
Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 8], {n, 0, 23}]
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
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