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

3,2

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).

Formula

a(n) = binomial(3n, n-3), n > 2. - Wesley Ivan Hurt, Feb 04 2014

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)

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 *)

Prog

(PARI) {a(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
(Sage) [binomial(3*n, n-3) for n in (3..30)] # G. C. Greubel, Mar 21 2019
(GAP) List([3..30], n-> Binomial(3*n, n-3)) # G. C. Greubel, Mar 21 2019

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).

Sequence in context: A094530 A157024 A051815 * A016223 A027142 A090816

Adjacent sequences: A004318 A004319 A004320 * A004322 A004323 A004324

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved