A002409 a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube. (Formerly M4939 N1668)

1, 14, 112, 672, 3360, 14784, 59136, 219648, 768768, 2562560, 8200192, 25346048, 76038144, 222265344, 635043840, 1778122752, 4889837568, 13231325184, 35283533824, 92851404800, 241413652480, 620777963520, 1580162088960

Offset

0,2

Comments

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>5, a(n-6) is equal to the number of (n+6)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

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).

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)

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

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 *)

Prog

(Magma) [2^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

Crossrefs

First differences are in A006976.

Cf. A000079, A001787, A001788, A001789, A003472, A054849, A054851, A082140.

a(n) = A038207(n+6,6).

Sequence in context: A213348 A341368 A004408 * A155655 A007817 A285147

Adjacent sequences: A002406 A002407 A002408 * A002410 A002411 A002412

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Henry Bottomley and James A. Sellers, Apr 15 2000

Typo in definition corrected by Zerinvary Lajos, Jun 16 2008

Status

approved