A242798 Expansion of -x*d(log((1-x*(2/sqrt(3*x)) * sin((1/3) * arcsin(sqrt(27*x/4))))))/dx.

0, 1, 3, 13, 67, 376, 2211, 13378, 82499, 515659, 3255628, 20714354, 132611491, 853226921, 5512508382, 35739673513, 232405291587, 1515159860388, 9900216370689, 64816750480666, 425097621975692, 2792332673312356, 18367642416256334, 120972943783673953

Offset: 0

Author

Vladimir Kruchinin, May 22 2014

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), #13.5.1.

Crossrefs

Cf. A001764, A174687.

Programs

Maple

ogf := n -> ((1 - sqrt(1 - 4*x))/(2*x))^n*x/(1 - x):
ser := n -> series(ogf(n), x, 46):
seq(coeff(ser(n), x, n), n=0..23); # Peter Luschny, Jan 25 2019

Mathematica

Table[n*Sum[Binomial[3*n - 2*k - 1, n - k]/(2*n - k), {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 21 2015 *)
Table[(n Binomial[3 n - 3, n - 1] HypergeometricPFQ[{1, 1 - 2 n, 1 - n}, {3/2 - (3 n)/2, 2 - (3 n)/2}, 1/4])/(2 n - 1), {n, 0, 23}] (* Peter Luschny, Jan 25 2019 *)

Maxima

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

PARI

a(n) = n*sum(k=1, n, binomial(3*n-2*k-1,n-k)/(2*n-k)); \\ Michel Marcus, Nov 12 2022

Formula

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

G.f.: x*(x*F'(x)+F(x))/(1-x*F(x)), where F(x) is g.f. of A001764.

D-finite with recurrence: 2*(n-1)*(2*n-1)*(91*n^3 - 531*n^2 + 962*n - 516)*a(n) = (2821*n^5 - 21921*n^4 + 62005*n^3 - 75435*n^2 + 33274*n - 24)*a(n-1) - (2821*n^5 - 21921*n^4 + 62005*n^3 - 75435*n^2 + 33274*n - 24)*a(n-2) + 3*(3*n - 8)*(3*n - 7)*(91*n^3 - 258*n^2 + 173*n + 6)*a(n-3). - Vaclav Kotesovec, Sep 21 2015

a(n) ~ 3^(3*n-1/2) / (7 * sqrt(Pi*n) * 4^n). - Vaclav Kotesovec, Sep 21 2015

From Peter Luschny, Jan 25 2019: (Start)

a(n) = (n/(2*n-1))*C(3*n-3, n-1)*(3F2)([1, 1-2*n, 1-n], [3/2-3*n/2, 2-3*n/2], 1/4).

a(n) = [x^n] (2/(1 + sqrt(1 - 4*x)))^n*(x/(1 - x)). (End)

Extensions

Name edited by Michel Marcus, Jan 26 2019