A129442 -id:A129442 - cp's OEIS frontend

A130655 Catalan transform of Catalan numbers C(n+1).

1, 2, 7, 28, 119, 524, 2363, 10844, 50446, 237280, 1126437, 5389916, 25967972, 125868952, 613385075, 3003586196, 14771851093, 72936101780, 361419276386, 1796837068400, 8960207761500

Offset: 0

Philippe Deléham, Jun 21 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A127632, A129442.

A130655 := proc(n)
    add(A106566(n,k)*A000108(k+1),k=0..n) ;
end proc: # R. J. Mathar, Mar 01 2015

CoefficientList[Series[2/(Sqrt[-1 + 2*Sqrt[1-4*x]] + Sqrt[1-4*x]),{x,0,20}],x] (* Vaclav Kotesovec, Jul 02 2015 *)

x='x+O('x^50); Vec(2/(sqrt(-1 + 2*sqrt(1-4*x)) + sqrt(1-4*x))) \\ G. C. Greubel, Mar 21 2017

a(n) = Sum_{k=0..n} A106566(n,k)*A000108(k+1).
Conjecture: 3*n*(n-2)*(n+2)*a(n) +4*(-10*n^3+21*n^2+7*n-15)*a(n-1) +16*(11*n^3-47*n^2+57*n-15)*a(n-2) -8*(2*n-5)*(4*n-9)*(4*n-7)*a(n-3)=0. - R. J. Mathar, Mar 01 2015
G.f.: (C(x*C(x))-1)/(x*C(x)), where C(x) is g.f. of Catalan numbers A000108. - Vladimir Kruchinin, Jul 02 2015
a(n) ~ 2^(4*n+3/2) / (sqrt(Pi) * n^(3/2) * 3^(n-1/2)). - Vaclav Kotesovec, Jul 02 2015

A368938 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x+x^2) ).

Original entry on oeis.org

1, 3, 14, 78, 479, 3129, 21332, 150057, 1081118, 7937589, 59174752, 446744610, 3408616155, 26242751046, 203615759472, 1590550846398, 12498584431503, 98731454253945, 783581338236326, 6245066800130298, 49961547869830135, 401076129627216180, 3229808459696023980

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A129442, A368934.

a(n) = sum(k=0, n\2, (-1)^k*binomial(n+k, k)*binomial(4*n-k+2, n-2*k))/(n+1);

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*(1-x+x^2))/x)

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(4*n-k+2,n-2*k).

A381826 G.f. A(x) satisfies A(x) = C(x) / (1 - x*A(x)^2), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 8, 41, 241, 1545, 10503, 74429, 543833, 4067510, 30985633, 239560975, 1874831287, 14823253892, 118222204539, 949963236834, 7683289712433, 62499664522578, 510992689465500, 4196824203859773, 34609480384100715, 286461380785102398, 2378954616256505177

Offset: 0

Seiichi Manyama, Mar 08 2025

nonn

Cf. A014137, A129442, A381827.

Cf. A000108.

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

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

D-finite with recurrence 12*n*(3*n+2)*(2*n+1)*(3*n+1)*a(n) +2*(-2365*n^4+2754*n^3-1799*n^2+834*n-144)*a(n-1) +2*(20215*n^4-89442*n^3+158117*n^2-135942*n+47592)*a(n-2) +(-181487*n^4+1469774*n^3-4524589*n^2+6309094*n-3370512)*a(n-3) +124*(n-3)*(2*n-7)*(1797*n^2-9448*n+12568)*a(n-4) -119164*(2*n-7)*(2*n-9)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Mar 10 2025

A381827 G.f. A(x) satisfies A(x) = C(x) / (1 - x*A(x)^3), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 10, 69, 562, 5042, 48100, 478547, 4908338, 51522174, 550758208, 5974753990, 65608248500, 727835313461, 8144965594184, 91834891588099, 1042244963201914, 11896871741939462, 136493661712053752, 1573151972820654218, 18205626549920314728, 211468167403628323318

Offset: 0

Seiichi Manyama, Mar 08 2025

nonn

Cf. A014137, A129442, A381826.

Cf. A000108, A381782.

a(n) = sum(k=0, n, binomial(3*n-k+1, k)*binomial(4*n-4*k, n-k)/(3*n-k+1));

a(n) = Sum_{k=0..n} binomial(3*n-k+1,k) * binomial(4*n-4*k,n-k)/(3*n-k+1).

cp's OEIS Frontend

A130655 Catalan transform of Catalan numbers C(n+1).

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A368938 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x+x^2) ).

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A381826 G.f. A(x) satisfies A(x) = C(x) / (1 - x*A(x)^2), where C(x) is the g.f. of A000108.

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A381827 G.f. A(x) satisfies A(x) = C(x) / (1 - x*A(x)^3), where C(x) is the g.f. of A000108.

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