A270489 Sum_{k=0..n} ((binomial(3*k,k)*binomial(2*n-k,n))/(2*k+1)).

1, 3, 12, 54, 265, 1401, 7903, 47088, 293319, 1892440, 12548041, 84988566, 585314652, 4085026386, 28820064810, 205156454376, 1471492171068, 10622954509803, 77122189800121, 562684397212060, 4123449352097229, 30336562360256955

Offset: 0

Author

Vladimir Kruchinin, Mar 18 2016

Keywords

• nonn

Links

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

Crossrefs

Cf. A000108, A001764, A092392.

Programs

Mathematica

Table[Sum[Binomial[3*k,k]*Binomial[2*n-k,n]/(2*k+1), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 18 2016 *)

Maxima

taylor((sqrt(2)*sin(asin((3^(3/2)*sqrt(1-sqrt(1-4*x)))/2^(3/2))/3)*
sqrt(1-sqrt(1-4*x)))/(sqrt(3)*(1-x/(1-(1-sqrt(1-4*x))/2)^2))/x,x,0,20);

PARI

for(n=0,25, print1(sum(k=0,n, binomial(3*k,k)*binomial(2*n-k,n)/(2*k+1)), ", ")) \\ G. C. Greubel, Jun 05 2017

Formula

G.f.: P(C(x))/(1-x/(1-C(x)))^2/x, where C(x)=(1-sqrt(1-4*x))/2, P(x)/x is g.f. of A001764.

Recurrence: 46*(n-2)*(n-1)*n*(2*n + 1)*(133*n - 239)*a(n) = 5*(n-2)*(n-1)*(38969*n^3 - 108996*n^2 + 78733*n - 18774)*a(n-1) - 4*(n-2)*(242858*n^4 - 1286417*n^3 + 2496793*n^2 - 2103937*n + 643755)*a(n-2) + 36*(3*n - 7)*(3*n - 5)*(6*n - 13)*(6*n - 11)*(133*n - 106)*a(n-3). - Vaclav Kotesovec, Mar 18 2016

a(n) ~ 3^(6*n + 7/2) / (19^(3/2) * sqrt(Pi) * 2^(2*n+2) * 23^(n - 1/2) * n^(3/2)). - Vaclav Kotesovec, Mar 18 2016