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6th binomial transform of (1,0,1,0,1,.....), A059841.
8

%I #27 Sep 08 2022 08:45:09

%S 1,6,37,234,1513,9966,66637,450834,3077713,21153366,146120437,

%T 1013077434,7042713913,49054856766,342163294237,2389039544034,

%U 16692759230113,116696726720166,816114147588037,5708984335850634

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

%C Binomial transform of A081187.

%C 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

%H Vincenzo Librandi, <a href="/A081188/b081188.txt">Table of n, a(n) for n = 0..200</a>

%H Takao Komatsu, <a href="https://doi.org/10.22436/jnsa.012.12.05">Some recurrence relations of poly-Cauchy numbers</a>, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-35).

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

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

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

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

%t CoefficientList[Series[(1 - 6 x) / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)

%t LinearRecurrence[{12,-35},{1,6},30] (* _Harvey P. Dale_, Mar 24 2016 *)

%o (Magma) [5^n/2+7^n/2: n in [0..25]]; // _Vincenzo Librandi_, Aug 07 2013

%o (PARI) a(n)=5^n/2+7^n/2 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A007582, A081186, A081189.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Mar 11 2003