A054851 a(n) = 2^(n-7)*binomial(n,7). Number of 7D hypercubes in an n-dimensional hypercube.

1, 16, 144, 960, 5280, 25344, 109824, 439296, 1647360, 5857280, 19914752, 65175552, 206389248, 635043840, 1905131520, 5588385792, 16066609152, 45364543488, 126012620800, 344876646400, 931166945280, 2483111854080

Offset: 7

Author

Henry Bottomley, Apr 14 2000

Keywords

• nonn
• easy

Comments

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>6, a(n) is equal to the number of (n+7)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 7..1000
• Milan Janjic, Two Enumerative Functions.
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792, 1024,-256).

Crossrefs

Cf. A000079, A001787, A001788, A001789, A003472, A054849, A002409.

a(n) = A038207(n,7).

Programs

GAP

List([7..30], n-> 2^(n-7)*Binomial(n,7)); # G. C. Greubel, Aug 27 2019

Magma

[2^(n-7)*Binomial(n,7): n in [7..30]]; // G. C. Greubel, Aug 27 2019

Maple

seq(binomial(n+7,7)*2^n,n=0..21); # Zerinvary Lajos, Jun 23 2008

Mathematica

Table[2^(n-7)*Binomial[n,7], {n,7,30}] (* G. C. Greubel, Aug 27 2019 *)

PARI

vector(23, n, 2^(n-1)*binomial(n+6, 7)) \\ G. C. Greubel, Aug 27 2019

Formula

a(n) = 2*a(n-1) + A002409(n-1).

a(n+8) = A082141(n+1)/2.

G.f.: x^7/(1-2*x)^8. - Colin Barker, Sep 04 2012

a(n) = Sum_{i=7..n} binomial(i,7)*binomial(n,i). Example: for n=11, a(11) = 1*330 + 8*165 + 36*55 + 120*11 + 330*1 = 5280. - Bruno Berselli, Mar 23 2018

From Amiram Eldar, Jan 06 2022: (Start)

Sum_{n>=7} 1/a(n) = 14*log(2) - 259/30.

Sum_{n>=7} (-1)^(n+1)/a(n) = 10206*log(3/2) - 124117/30. (End)

Extensions

More terms from James Sellers, Apr 15 2000