A004334 Binomial coefficient C(4n,n-4).

1, 20, 276, 3276, 35960, 376992, 3838380, 38320568, 377348994, 3679075400, 35607051480, 342700125300, 3284214703056, 31368725759168, 298824321028320, 2840671544105280, 26958221130508525, 255485622301674660

Offset: 4

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

• Vincenzo Librandi, Table of n, a(n) for n = 4..1000
• 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].
• 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), A004321 (k=3), this sequence (k=4), A004347 (k=5), A004361 (k=6), A004375 (k=7), A004389 (k=8), A281580 (k=9).

Programs

GAP

List([4..30], n-> Binomial(4*n,n-4)); # G. C. Greubel, Mar 21 2019

Magma

[Binomial(4*n, n-4): n in [4..30]]; // Vincenzo Librandi, Feb 01 2017

Mathematica

Table[Binomial[4n, n-4], {n,4,30}] (* Vincenzo Librandi, Feb 01 2017 *)

PARI

a(n)=binomial(4*n,n-4) \\ Charles R Greathouse IV, Feb 01 2017

Sage

[binomial(4*n,n-4) for n in (4..30)] # G. C. Greubel, Mar 21 2019

Formula

From Ilya Gutkovskiy, Jan 31 2017: (Start)

E.g.f.: (1/24)*x^4*3F3(17/4,9/2,19/4; 17/3,6,19/3; 256*x/27).

a(n) ~ 2^(8*n+1/2)/(sqrt(Pi*n)*3^(3*n+9/2)). (End)

D-finite with recurrence -3*(3*n+2)*(n-4)*(3*n+4)*(n+1)*a(n) +8*n*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 19 2025