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%I #25 Sep 08 2022 08:45:09
%S 0,1,6,29,130,561,2366,9829,40410,164921,669526,2707629,10919090,
%T 43942081,176565486,708653429,2841788170,11388676041,45619274246,
%U 182670807229,731264359650,2926800830801,11712433499806,46865424529029
%N Generalized Poly-Bernoulli numbers.
%C Binomial transform of A027649. Inverse binomial transform of A081675
%C With offset 1, partial sums of A085350. - _Paul Barry_, Jun 24 2003
%C Number of walks of length 2n+2 between two nodes at distance 4 in the cycle graph C_12. - _Herbert Kociemba_, Jul 05 2004
%H Vincenzo Librandi, <a href="/A081674/b081674.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-19,12).
%F a(n) = ((4^(n+1) - 1)/3 - 3^n)/2 = (4*4^n - 3*3^n - 1)/6.
%F a(n) = (A002450(n+1) + A000244(n))/2.
%F G.f.: x*(1-2*x)/((1-x)*(1-3*x)*(1-4*x)).
%t Join[{a=0,b=1},Table[c=7*b-12*a-1;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 19 2011 *)
%t CoefficientList[Series[(x(1-2x))/((1-x)(1-3x)(1-4x)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-19,12},{0,1,6},30] (* _Harvey P. Dale_, Nov 28 2018 *)
%o (Magma) [((4^(n+1)-1)/3-3^n)/2: n in [0..30]]; // _Vincenzo Librandi_, Jul 17 2011
%K easy,nonn
%O 0,3
%A _Paul Barry_, Mar 28 2003
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