A188686 Binomial transform of the sequence of binomial(3n,n).

1, 4, 22, 139, 934, 6484, 45931, 329893, 2393470, 17499892, 128732992, 951674398, 7064138779, 52616241370, 393052285291, 2943582912904, 22093111508686, 166141033332448, 1251528633163264, 9442096410241438, 71333250226656784

Offset: 0

Author

Emanuele Munarini, Apr 08 2011

Keywords

• nonn
• easy

Comments

Binomial transform of A005809.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A005809, A359643.

Programs

Mathematica

Table[Sum[Binomial[n,k]Binomial[3k,k],{k,0,n}],{n,0,22}]

Maxima

makelist(sum(binomial(n,k)*binomial(3*k,k),k,0,n),n,0,20);

Formula

G.f.: 2*cos((1/3)*arcsin(3/2*sqrt(3x/(1-x))))/sqrt(4-35x+31x^2).

D-finite recurrence: 2*n*(2*n-1)*a(n) = (39*n^2-43*n+12)*a(n-1) - 2*(n-1)*(33*n-34)*a(n-2) + 31*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 31^(n+1/2)/(6*4^n*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012

a(n) = [x^n] (1 + 4*x + 3*x^2 + x^3)^n. - Ilya Gutkovskiy, Apr 17 2025