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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (7, -12).
Programs
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Magma
[2*4^n-3^n: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011
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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).
Comments