A081201 7th binomial transform of (0,1,0,1,0,1,....), A000035.

0, 1, 14, 148, 1400, 12496, 107744, 908608, 7548800, 62070016, 506637824, 4113568768, 33271347200, 268347559936, 2159841173504, 17357093552128, 139326933401600, 1117436577120256, 8956419276406784, 71752914167922688, 574632673083392000, 4600717543107198976

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• nonn
• easy
• changed

Comments

Binomial transform of A081200.

Conjecture (verified up to a(8)): Number of collinear 6-tuples of points in a 6 X 6 X 6 X... n-dimensional cubic grid. [R. H. Hardin, May 23 2010]

From Wolfdieter Lang, Jul 17 2017: (Start)

For a combinatorial interpretation of a(n) with special 8-letter words of length n see the comment in A081200 on the 7-letter analog.

The binomial transform of {a(n)}_{n >= 0} is A081202, the 9-letter analog.

(End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (14,-48).

Crossrefs

Cf. A000035, A016169, A081200, A081202.

Apart from offset same as A016170.

Programs

Magma

[(8^n-6^n)/2: n in [0..30]]; // Vincenzo Librandi, Aug 07 2013

Mathematica

CoefficientList[Series[x/((1 - 6 x) (1 - 8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{14,-48},{0,1},30] (* Harvey P. Dale, Oct 24 2022 *)

SageMath

A081201=BinaryRecurrenceSequence(14,-48,0,1)
[A081201(n) for n in range(41)] # G. C. Greubel, Nov 10 2024

Formula

a(n) = 14*a(n-1) - 48*a(n-2) with n>1, a(0)=0, a(1)=1.

G.f.: x/((1-6*x)*(1-8*x)).

a(n) = (1/2)*(8^n - 6^n).

E.g.f.: exp(7*x)*sinh(x). - G. C. Greubel, Nov 10 2024

Extensions

Name clarified by Pontus von Brömssen, Nov 11 2020