A054849 a(n) = 2^(n-5)*binomial(n,5). Number of 5D hypercubes in an n-dimensional hypercube.

1, 12, 84, 448, 2016, 8064, 29568, 101376, 329472, 1025024, 3075072, 8945664, 25346048, 70189056, 190513152, 508035072, 1333592064, 3451650048, 8820883456, 22284337152, 55710842880, 137950658560, 338606161920

Offset: 5

Author

Henry Bottomley, Apr 14 2000

Keywords

• easy
• nonn

Comments

With 5 leading zeros, binomial transform of binomial(n,5). - Paul Barry, Apr 10 2003

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>4, a(n) is equal to the number of (n+5)-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 = 5..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 (12, -60, 160, -240, 192, -64).

Crossrefs

Cf. A000079, A001787, A001788, A001789, A003472, A002409, A054851.

a(n) = A038207(n,5).

Equals 2 * A082139. First differences are in A006975.

Programs

GAP

List([5..30], n-> 2^(n-5)*Binomial(n,5)); # G. C. Greubel, Aug 27 2019

Magma

[2^(n-5)*Binomial(n,5): n in [5..30]]; // G. C. Greubel, Aug 27 2019

Maple

seq(binomial(n+5,5)*2^n,n=0..22); # Zerinvary Lajos, Jun 13 2008

Mathematica

Table[2^(n-5)*Binomial[n,5], {n,5,30}] (* G. C. Greubel, Aug 27 2019 *)

PARI

vector(25, n, 2^(n-1)*binomial(n+4,5)) \\ G. C. Greubel, Aug 27 2019

Sage

[lucas_number2(n, 2, 0)*binomial(n,5)/32 for n in range(5, 28)] # Zerinvary Lajos, Mar 10 2009

Formula

a(n) = 2*a(n-1) + A003472(n-1).

From Paul Barry, Apr 10 2003: (Start)

O.g.f.: x^5/(1-2*x)^6.

E.g.f.: exp(2*x)*(x^5/5!) (with 5 leading zeros). (End)

a(n) = Sum_{i=5..n} binomial(i,5)*binomial(n,i). Example: for n=8, a(8) = 1*56 + 6*28 + 21*8 + 56*1 = 448. - Bruno Berselli, Mar 23 2018

From Amiram Eldar, Jan 06 2022: (Start)

Sum_{n>=5} 1/a(n) = 10*log(2) - 35/6.

Sum_{n>=5} (-1)^(n+1)/a(n) = 810*log(3/2) - 655/2. (End)

Extensions

More terms from James Sellers, Apr 15 2000