A081188 6th binomial transform of (1,0,1,0,1,.....), A059841.

1, 6, 37, 234, 1513, 9966, 66637, 450834, 3077713, 21153366, 146120437, 1013077434, 7042713913, 49054856766, 342163294237, 2389039544034, 16692759230113, 116696726720166, 816114147588037, 5708984335850634

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A081187.

a(n) is also the number of words of length n over an alphabet of seven letters, of which a chosen one appears an even number of times. See a comment in A007582, also for the crossrefs. for the 1- to 11- letter word cases. - Wolfdieter Lang, Jul 17 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
• Index entries for linear recurrences with constant coefficients, signature (12,-35).

Crossrefs

Cf. A007582, A081186, A081189.

Programs

Magma

[5^n/2+7^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

Mathematica

CoefficientList[Series[(1 - 6 x) / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12,-35},{1,6},30] (* Harvey P. Dale, Mar 24 2016 *)

PARI

a(n)=5^n/2+7^n/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 12*a(n-1) -35*a(n-2), a(0)=1, a(1)=6.

G.f.: (1-6*x)/((1-5*x)*(1-7*x)).

E.g.f.: exp(6*x)*cosh(x).

a(n) = 5^n/2 + 7^n/2.