A004319 a(n) = binomial(3*n, n - 1).

1, 6, 36, 220, 1365, 8568, 54264, 346104, 2220075, 14307150, 92561040, 600805296, 3910797436, 25518731280, 166871334960, 1093260079344, 7174519270695, 47153358767970, 310325523515700, 2044802197953900, 13488561475572645, 89067326568860640, 588671286046028640

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• Milan Janjic, Two Enumerative Functions.
• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Crossrefs

Cf. A045721 (k=1), A025174 (k=2), A236194 (k=4), A013698 (k=5), A165817 (k=-1), A117671 (k=-2).

Cf. A000139, A001764, A005809, A006013, A236194, A001791, A004331, A004343, A004356, A004369, A004382.

Programs

Maple

A004319 := proc(n)
binomial(3*n,n-1);
end proc: # R. J. Mathar, Aug 10 2015

Mathematica

Table[Binomial[3n, n - 1], {n, 20}] (* Harvey P. Dale, Sep 21 2011 *)

Maxima

a(n):=sum((binomial(3*i-1,2*i-1)*binomial(3*n-3*i-3,2*n-2*i-2))/(2*n-2*i-1),i,1,n-1)/2; /* Vladimir Kruchinin, May 15 2013 */

PARI

vector(30, n, binomial(3*n, n-1)) \\ Altug Alkan, Nov 04 2015

Formula

G.f.: (g-1)/(1-3*z*g^2), where g = g(z) is given by g = 1 + z*g^3, g(0) = 1, i.e. (in Maple notation), g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003

a(n) = Sum_{i=0..n-1} binomial(i+2*n, i). - Ralf Stephan, Jun 03 2005

D-finite with recurrence -2*(2*n+1)*(n-1)*a(n) + 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013

a(n) = (1/2) * Sum_{i=1..n-1} binomial(3*i - 1, 2*i - 1)*binomial(3*n - 3*i - 3, 2*n - 2*i - 2)/(2*n - 2*i - 1). - Vladimir Kruchinin, May 15 2013

G.f.: x*hypergeom2F1(5/3, 4/3; 5/2; 27x/4). - R. J. Mathar, Aug 10 2015

a(n) = n*A001764(n). - R. J. Mathar, Aug 10 2015

From Peter Bala, Nov 04 2015: (Start)

With offset 0, the o.g.f. equals f(x)*g(x)^3, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k, n). See the cross-references. (End)

G.f.: cos(t)/(2*sqrt(1 - (27*x)/4)) - sin(t)/(sqrt(3)*sqrt(x)), where t = arcsin((sqrt(27*x))/2)/3. - Vladimir Kruchinin, May 13 2016

a(n) = [x^(2*n+1)] 1/(1 - x)^n. - Ilya Gutkovskiy, Oct 10 2017

a(n) = binomial(n+1, 2) * A000139(n). - F. Chapoton, Feb 23 2024