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

0,3

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).

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A081278 A054146 A172062 * A173413 A008549 A345031

Adjacent sequences: A081671 A081672 A081673 * A081675 A081676 A081677

Keyword

easy,nonn

Author

Paul Barry, Mar 28 2003

Status

approved