A002409 a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube.
1, 14, 112, 672, 3360, 14784, 59136, 219648, 768768, 2562560, 8200192, 25346048, 76038144, 222265344, 635043840, 1778122752, 4889837568, 13231325184, 35283533824, 92851404800, 241413652480, 620777963520, 1580162088960
Offset: 0
References
- 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
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
- Herbert Izbicki, Über Unterbaeume eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
- Herbert Izbicki, Über Unterbaeume 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.
- 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.
- Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).
Crossrefs
Programs
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Magma
[2^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011
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Maple
A002409:=-1/(2*z-1)**7; # Simon Plouffe in his 1992 dissertation seq(binomial(n+6,6)*2^n,n=0..22); # Zerinvary Lajos, Jun 16 2008
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Mathematica
CoefficientList[Series[1/(1-2x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {14,-84,280,-560,672,-448,128},{1,14,112,672,3360,14784,59136},40] (* Harvey P. Dale, Jan 24 2022 *)
Formula
G.f.: 1/(1-2*x)^7.
a(n) = 2*a(n-1) + A054849(n-1).
For n>0, a(n) = 2*A082140(n).
a(n) = Sum_{i=6..n+6} binomial(i,6)*binomial(n+6,i). Example: for n=5, a(5) = 1*462 + 7*330 + 28*165 + 84*55 + 210*11 + 462*1 = 14784. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 47/5 - 12*log(2).
Sum_{n>=0} (-1)^n/a(n) = 2916*log(3/2) - 5907/5. (End)
n*a(n) +2*(-n-6)*a(n-1)=0. - R. J. Mathar, Jul 22 2025
Extensions
More terms from Henry Bottomley and James Sellers, Apr 15 2000
Typo in definition corrected by Zerinvary Lajos, Jun 16 2008
Comments