A081674 Generalized Poly-Bernoulli numbers.

0, 1, 6, 29, 130, 561, 2366, 9829, 40410, 164921, 669526, 2707629, 10919090, 43942081, 176565486, 708653429, 2841788170, 11388676041, 45619274246, 182670807229, 731264359650, 2926800830801, 11712433499806, 46865424529029, 187508769705530, 750176293590361, 3001128818666166

Offset: 0

Author

Paul Barry, Mar 28 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A027649. Inverse binomial transform of A081675.

With offset 1, partial sums of A085350. - Paul Barry, Jun 24 2003

Number of walks of length 2n+2 between two nodes at distance 4 in the cycle graph C_12. - Herbert Kociemba, Jul 05 2004

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-19,12).

Crossrefs

Cf. A000244, A002450, A027649, A081675, A085350.

Programs

Magma

[((4^(n+1)-1)/3-3^n)/2: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

Mathematica

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 *)
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 *)

Formula

a(n) = ((4^(n+1) - 1)/3 - 3^n)/2 = (4*4^n - 3*3^n - 1)/6.

a(n) = (A002450(n+1) + A000244(n))/2.

G.f.: x*(1-2*x)/((1-x)*(1-3*x)*(1-4*x)).

From Elmo R. Oliveira, Aug 29 2024: (Start)

E.g.f.: exp(x)*(4*exp(3*x) - 3*exp(2*x) - 1)/6.

a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3) for n > 2. (End)