A085350 Binomial transform of poly-Bernoulli numbers A027649.

1, 5, 23, 101, 431, 1805, 7463, 30581, 124511, 504605, 2038103, 8211461, 33022991, 132623405, 532087943, 2133134741, 8546887871, 34230598205, 137051532983, 548593552421, 2195536471151, 8785632669005, 35152991029223

Offset

0,2

Comments

Binomial transform is A085351.

a(n) mod 10 = period 4:repeat 1,5,3,1 = A132400. - Paul Curtz, Nov 13 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (7, -12).

Formula

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

E.g.f.: 2exp(4x) - exp(3x).

a(n) = 2*4^n-3^n.

From Paul Curtz, Nov 13 2009: (Start)

a(n) = 4*a(n-1) + 9*a(n-2) - 36*a(n-3);

a(n) = 4*a(n-1) + 3^(n-1), both like A005061 (note for A005061 dual formula a(n) = 3*a(n-1) + 4^(n-1) = 3*a(n-1) + A000302(n-1)).

a(n) = 3*a(n-1) + 2^(2n+1) = 3*a(n-1) + A004171(n).

a(n) = A005061(n) + A000302(n).

b(n) = mix(A005061, A085350) = 0,1,1,5,7,23,... = differences of (A167762 = 0,0,1,2,7,14,37,...); b(n) differences = A167784. (End)

Mathematica

LinearRecurrence[{4, 9, -36}, {1, 5, 23}, 30] (* Harvey P. Dale, Nov 30 2011 *)
LinearRecurrence[{7, -12}, {1, 5}, 23] (* Ray Chandler, Aug 03 2015 *)

Prog

(Magma) [2*4^n-3^n: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

Crossrefs

a(n-1) = A080643(n)/2 = A081674(n+1) - A081674(n).

Cf. A000244 (3^n).

Sequence in context: A034958 A229008 A274322 * A113443 A124999 A258431

Adjacent sequences: A085347 A085348 A085349 * A085351 A085352 A085353

Keyword

easy,nonn

Author

Paul Barry, Jun 24 2003

Status

approved