A081199 5th binomial transform of (0,1,0,1,...), A000035.

0, 1, 10, 76, 520, 3376, 21280, 131776, 807040, 4907776, 29708800, 179301376, 1080002560, 6496792576, 39047864320, 234555621376, 1408407470080, 8454739787776, 50745618595840, 304542431051776, 1827529464217600, 10966276296933376, 65802055828111360, 394829927154712576

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• nonn
• easy
• changed

Comments

Binomial transform of A005059.

Conjecture (verified up to a(9)): Number of collinear 4-tuples of points in a 4 X 4 X 4 X ... n-dimensional cubic grid. - R. H. Hardin, May 24 2010

a(n) is also the total number of words of length n, over an alphabet of six letters, of which one of them appears an odd number of times. See a Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5- and 7-letter case analogs see A131577, A003462, A005059 and A081200, respectively. - Wolfdieter Lang, Jul 16 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (10,-24).

Crossrefs

Cf. A000035, A003462, A005059, A006516, A081200 (binomial transform of a(n), and 7-letter case), A131577.

Apart from offset the same as A016149.

Programs

Magma

[6^n/2-4^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

Maple

seq(add(2^(2*n-k)*binomial(n,k)/2,k=1..n),n=0..20); # Zerinvary Lajos, Apr 18 2009

Mathematica

CoefficientList[Series[x / ((1 - 4 x) (1 - 6 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{10, -24}, {0, 1}, 21] (* Michael De Vlieger, Jul 16 2017 *)

Formula

a(n) = 10*a(n-1) - 24*a(n-2) with n>1, a(0)=0, a(1)=1.

G.f.: x/((1-4*x)*(1-6*x)).

a(n) = 6^n/2 - 4^n/2.

E.g.f.: exp(4*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jul 23 2024