A004321 Binomial coefficient C(3n, n-3).

1, 12, 105, 816, 5985, 42504, 296010, 2035800, 13884156, 94143280, 635745396, 4280561376, 28760021745, 192928249296, 1292706174900, 8654327655120, 57902201338905, 387221678682300, 2588713818544245

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 3..1209
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
• Milan Janjic, Two Enumerative Functions
• Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

Crossrefs

Cf. binomial(k*n, n-k): A000027 (k=1), A002694 (k=2), this sequence (k=3), A004334 (k=4), A004347 (k=5), A004361 (k=6), A004375 (k=7), A004389 (k=8), A281580 (k=9).

Programs

GAP

List([3..30], n-> Binomial(3*n,n-3)); # G. C. Greubel, Mar 21 2019

Magma

[Binomial(3*n,n-3): n in [3..30]]; // G. C. Greubel, Mar 21 2019

Maple

a:=n->sum(binomial(2*n-2,n+j)*binomial(n-1,n-j+1),j=0..n): seq(a(n), n=4..22); # Zerinvary Lajos, Jan 29 2007

Mathematica

Table[Binomial[3n, n-3], {n,3,30}] (* Wesley Ivan Hurt, Feb 04 2014 *)

PARI

{a(n) = binomial(3*n, n-3)}; \\ G. C. Greubel, Mar 21 2019

Sage

[binomial(3*n,n-3) for n in (3..30)] # G. C. Greubel, Mar 21 2019

Formula

From Ilya Gutkovskiy, Jan 31 2017: (Start)

E.g.f.: (1/6)*x^3*2F2(10/3,11/3; 5,11/2; 27*x/4).

a(n) ~ 3^(3*n+1/2)/(sqrt(Pi*n)*4^(n+2)). (End)

D-finite with recurrence -2*(2*n+3)*(n-3)*(n+1)*a(n) +3*n*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Jan 13 2025