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

Author

Paul Barry, Jun 24 2003

Keywords

• easy
• nonn

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).

Crossrefs

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

Cf. A000244 (3^n).

Programs

Magma

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

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 *)

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)