A001789 a(n) = binomial(n,3)*2^(n-3). (Formerly M4522 N1916)

1, 8, 40, 160, 560, 1792, 5376, 15360, 42240, 112640, 292864, 745472, 1863680, 4587520, 11141120, 26738688, 63504384, 149422080, 348651520, 807403520, 1857028096, 4244635648, 9646899200, 21810380800, 49073356800, 109924319232

Offset

3,2

Comments

Number of 3-dimensional cubes in n-dimensional hypercube. - Henry Bottomley, Apr 14 2000

With three leading zeros, this is the second binomial transform of (0,0,0,1,0,0,0,0,...). - Paul Barry, Mar 07 2003

With 3 leading zeros, binomial transform of C(n,3). - Paul Barry, Apr 10 2003

Let M=[1,0,i;0,1,0;i,0,1], i=sqrt(-1). Then 1/det(I-xM)=1/(1-2x)^4. - Paul Barry, Apr 27 2005

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>2, a(n+1) is equal to the number of (n+3)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

With offset 0, a(n) is the number of ways to separate [n] into four non-overlapping intervals (allowed to be empty) and then choose a subset from each interval. - Geoffrey Critzer, Feb 07 2009

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. 796.

Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 3..500

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].

H. J. Brothers, Pascal's Prism: Supplementary Material.

Herbert Izbicki, Über Unterbaüme eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.

Milan Janjic, Two Enumerative Functions.

Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.

C. W. Jones, J. C. P. Miller, J. F. C. Conn and R. C. Pankhurst, Tables of Chebyshev polynomials Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.

Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - From N. J. A. Sloane, Sep 04 2012

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Eric Weisstein's World of Mathematics, Hypercube.

Alina F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, Vol. 17 (2014), Article 14.10.3.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Formula

a(n) = 2*a(n-1) + A001788(n-2).

For n>0, a(n+3) = 2*A082138(n) = 8*A080930(n+1).

From Paul Barry, Apr 10 2003: (Start)

G.f. (with three leading zeros): x^3/(1-2*x)^4.

With three leading zeros, a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4), a(0)=a(1)=a(2)=0, a(3)=1.

E.g.f.: (x^3/3!)*exp(2*x) (with 3 leading zeros). (End)

a(n) = Sum_{i=3..n} binomial(i,3)*binomial(n,i). Example: for n=6, a(6) = 1*20 + 4*15 + 10*6 + 20*1 = 160. - Bruno Berselli, Mar 23 2018

From Amiram Eldar, Jan 06 2022: (Start)

Sum_{n>=3} 1/a(n) = 6*log(2) - 3.

Sum_{n>=3} (-1)^(n+1)/a(n) = 54*log(3/2) - 21. (End)

Maple

A001789:=1/(2*z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
seq(binomial(n+3, 3)*2^n, n=0..25); # Zerinvary Lajos, Jun 03 2008

Mathematica

Table[Binomial[n, 3]*2^(n-3), {n, 3, 30}] (* Stefan Steinerberger, Apr 18 2006 *)
LinearRecurrence[{8, -24, 32, -16}, {1, 8, 40, 160}, 30] (* Harvey P. Dale, Feb 10 2016 *)

Prog

(Haskell)
a001789 n = a007318 n 3 * 2 ^ (n - 3)
a001789_list = 1 : zipWith (+) (map (* 2) a001789_list) (drop 2 a001788_list)
-- Reinhard Zumkeller, Jul 12 2014
(PARI) a(n)=binomial(n, 3)<<(n-3) \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [Binomial(n, 3)*2^(n-3): n in [3..30]]; // G. C. Greubel, Aug 27 2019
(GAP) List([3..30], n-> Binomial(n, 3)*2^(n-3)); # G. C. Greubel, Aug 27 2019

Crossrefs

Cf. A000079, A001787, A001788, A003472, A007318, A082138, A080930.

a(n) = A038207(n,3).

Sequence in context: A341365 A004405 A284286 * A074412 A364619 A217375

Adjacent sequences: A001786 A001787 A001788 * A001790 A001791 A001792

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Apr 15 2000

More terms from Stefan Steinerberger, Apr 18 2006

Formula fixed by Reinhard Zumkeller, Jul 12 2014

Status

approved