A068764 -id:A068764 - cp's OEIS frontend

A068771 Generalized Catalan numbers 9xA(x)^2 -A(x) +1 -8*x=0.

1, 1, 18, 333, 6318, 122634, 2429028, 48974949, 1002875094, 20814628158, 437088964860, 9272342710962, 198456435657036, 4280758166952756, 92972201833888200, 2031520673763657621, 44630859892110807654

Offset: 0

Wolfdieter Lang, Mar 04 2002

Fung Lam, Table of n, a(n) for n = 0..725

Cf. A000108, A068764-A068770, A068772, A025227-A025230.

a[n_] := (288 (2 - n) a[n - 2] + 18 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}](* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-36*x*(1-8*x)])/(18*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n) = (9^n) * p(n, -8/9) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 9*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-36*x*(1-8*x)))/(18*x).
Recurrence: (n+1)*a(n) = 288*(2-n)*a(n-2) + 18*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(2) * 24^n / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

A364437 G.f. satisfies A(x) = 1 - x(1 - 2A(x)^3).

Original entry on oeis.org

1, 1, 6, 42, 326, 2712, 23676, 214068, 1987488, 18838464, 181548960, 1773566208, 17523740592, 174814263088, 1758342057504, 17812729393248, 181581358338528, 1861259423846400, 19172185074938112, 198354225907274496, 2060279149742042112

Offset: 0

Seiichi Manyama, Jul 24 2023

nonn

Cf. A068764.

A364437 := proc(n)
    (-1)^n*add((-2)^k* binomial(n,k) * binomial(3*k+1,n) / (3*k+1),k=0..n) ;
end proc:
seq(A364437(n),n=0..70); # R. J. Mathar, Jul 25 2023

a(n) = (-1)^n*sum(k=0, n, (-2)^k*binomial(n, k)*binomial(3*k+1, n)/(3*k+1));

a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(3*k+1,n) / (3*k+1).

D-finite with recurrence n*(2*n+1)*a(n) +3*(-11*n^2+14*n-4)*a(n-1) +27*(5*n-7) *(n-2)*a(n-2) -27*(7*n-16)*(n-3)*a(n-3) +81*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 25 2023

A068765 Generalized Catalan numbers 3xA(x)^2 -A(x) +1 -2*x = 0.

Original entry on oeis.org

1, 1, 6, 39, 270, 1962, 14796, 114831, 911574, 7368894, 60457428, 502162902, 4214515212, 35686162548, 304491863448, 2615468845311, 22598114065254, 196269877811574, 1712578870493316, 15005719955119698

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n)=K(3,3; n)/3 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..1000

Cf. A000108, A068764, A068766-72, A025227-30.

CoefficientList[Series[(1-Sqrt[1-12*x*(1-2*x)])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n)=(3^n)*p(n, -2/3) with the row polynomials p(n, x) defined from array A068763.
a(n+1)= 3*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-12*x*(1-2*x)))/(6*x).
Recurrence: (n+1)*a(n) = 24*(2-n)*a(n-2) + 6*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(6+6*sqrt(3)) * (6+2*sqrt(3))^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

A294642 a(n) = n! * [x^n] exp(nx)BesselI(1,2sqrt(2)x)/(sqrt(2)*x).

Original entry on oeis.org

1, 1, 6, 45, 456, 5825, 89896, 1627437, 33822944, 793783233, 20765009344, 599157626925, 18904594000128, 647524807918209, 23929038677825152, 948995910652193325, 40203601321988822528, 1812025020244371552897, 86577002960871477916672, 4371100278517527047687213

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000108, A001003, A025235, A068764, A071356, A151374, A247496, A292716.

Simplify[Table[n! SeriesCoefficient[Exp[n x] BesselI[1, 2 Sqrt[2] x]/(Sqrt[2] x), {x, 0, n}], {n, 0, 19}]]
Table[SeriesCoefficient[(1 - n x - Sqrt[1 - 2 n x + (n^2 - 8) x^2])/(4 x^2), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-2 x^2, 1 - n x, {i, 1, n}]), {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[2^k n^(n - 2 k) Binomial[n, 2 k] CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 1, 19}]]
Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {2}, 8/n^2], {n, 1, 19}]]

a(n) = [x^n] (1 - n*x - sqrt(1 - 2*n*x + (n^2 - 8)*x^2))/(4*x^2).
a(n) = [x^n] 1/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - ...))))), a continued fraction.
a(n) = Sum_{k=0..floor(n/2)} 2^k*n^(n-2*k)*binomial(n,2*k)*A000108(k).
a(n) = n^n*2F1(1/2-n/2,-n/2; 2; 8/n^2).
a(n) ~ c * n^n, where c = BesselI(1, 2*sqrt(2))/sqrt(2) = 2.3948330992734... - Vaclav Kotesovec, Nov 06 2017

cp's OEIS Frontend

A068771 Generalized Catalan numbers 9xA(x)^2 -A(x) +1 -8*x=0.

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A364437 G.f. satisfies A(x) = 1 - x(1 - 2A(x)^3).

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A068765 Generalized Catalan numbers 3xA(x)^2 -A(x) +1 -2*x = 0.

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Mathematica

Formula

A294642 a(n) = n! * [x^n] exp(nx)BesselI(1,2sqrt(2)x)/(sqrt(2)*x).

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cp's OEIS Frontend

A068771 Generalized Catalan numbers 9*x*A(x)^2 -A(x) +1 -8*x=0.

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Author

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Mathematica

Formula

A364437 G.f. satisfies A(x) = 1 - x*(1 - 2*A(x)^3).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Formula

A068765 Generalized Catalan numbers 3*x*A(x)^2 -A(x) +1 -2*x = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A294642 a(n) = n! * [x^n] exp(n*x)*BesselI(1,2*sqrt(2)*x)/(sqrt(2)*x).

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Mathematica

Formula

A068771 Generalized Catalan numbers 9xA(x)^2 -A(x) +1 -8*x=0.

A364437 G.f. satisfies A(x) = 1 - x(1 - 2A(x)^3).

A068765 Generalized Catalan numbers 3xA(x)^2 -A(x) +1 -2*x = 0.

A294642 a(n) = n! * [x^n] exp(nx)BesselI(1,2sqrt(2)x)/(sqrt(2)*x).