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Binomial coefficient C(3n, n-3).
6

%I #38 Mar 22 2023 20:34:06

%S 1,12,105,816,5985,42504,296010,2035800,13884156,94143280,635745396,

%T 4280561376,28760021745,192928249296,1292706174900,8654327655120,

%U 57902201338905,387221678682300,2588713818544245

%N Binomial coefficient C(3n, n-3).

%D 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.

%H Michael De Vlieger, <a href="/A004321/b004321.txt">Table of n, a(n) for n = 3..1209</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, <a href="https://arxiv.org/abs/2303.10986">Refined product formulas for Tamari intervals</a>, arXiv:2303.10986 [math.CO], 2023.

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>

%H Daniel W. Stasiuk, <a href="http://hdl.handle.net/10388/11865">An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads</a>, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

%F a(n) = binomial(3n, n-3), n > 2. - _Wesley Ivan Hurt_, Feb 04 2014

%F From _Ilya Gutkovskiy_, Jan 31 2017: (Start)

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

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

%p 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

%t Table[Binomial[3n, n-3], {n,3,30}] (* _Wesley Ivan Hurt_, Feb 04 2014 *)

%o (PARI) {a(n) = binomial(3*n, n-3)}; \\ _G. C. Greubel_, Mar 21 2019

%o (Magma) [Binomial(3*n,n-3): n in [3..30]]; // _G. C. Greubel_, Mar 21 2019

%o (Sage) [binomial(3*n,n-3) for n in (3..30)] # _G. C. Greubel_, Mar 21 2019

%o (GAP) List([3..30], n-> Binomial(3*n,n-3)) # _G. C. Greubel_, Mar 21 2019

%Y 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).

%K nonn,easy

%O 3,2

%A _N. J. A. Sloane_