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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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).

Crossrefs

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

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

Programs

GAP

List([3..30], n-> Binomial(n,3)*2^(n-3)); # G. C. Greubel, Aug 27 2019

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

Magma

[Binomial(n,3)*2^(n-3): n in [3..30]]; // G. C. Greubel, Aug 27 2019

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 *)

PARI

a(n)=binomial(n,3)<<(n-3) \\ Charles R Greathouse IV, Sep 24 2015

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)

Extensions

More terms from James Sellers, Apr 15 2000

More terms from Stefan Steinerberger, Apr 18 2006

Formula fixed by Reinhard Zumkeller, Jul 12 2014