cp's OEIS Frontend

A085350 Binomial transform of poly-Bernoulli numbers A027649.

%I A085350 #37 Aug 24 2024 18:20:37
%S A085350 1,5,23,101,431,1805,7463,30581,124511,504605,2038103,8211461,
%T A085350 33022991,132623405,532087943,2133134741,8546887871,34230598205,
%U A085350 137051532983,548593552421,2195536471151,8785632669005,35152991029223
%N A085350 Binomial transform of poly-Bernoulli numbers A027649.
%C A085350 Binomial transform is A085351.
%C A085350 a(n) mod 10 = period 4:repeat 1,5,3,1 = A132400. - _Paul Curtz_, Nov 13 2009
%H A085350 Vincenzo Librandi, <a href="/A085350/b085350.txt">Table of n, a(n) for n = 0..1000</a>
%H A085350 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7, -12).
%F A085350 G.f.: (1-2x)/((1-3x)(1-4x)).
%F A085350 E.g.f.: 2exp(4x) - exp(3x).
%F A085350 a(n) = 2*4^n-3^n.
%F A085350 From _Paul Curtz_, Nov 13 2009: (Start)
%F A085350 a(n) = 4*a(n-1) + 9*a(n-2) - 36*a(n-3);
%F A085350 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)).
%F A085350 a(n) = 3*a(n-1) + 2^(2n+1) = 3*a(n-1) + A004171(n).
%F A085350 a(n) = A005061(n) + A000302(n).
%F A085350 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)
%t A085350 LinearRecurrence[{4,9,-36},{1,5,23},30] (* _Harvey P. Dale_, Nov 30 2011 *)
%t A085350 LinearRecurrence[{7, -12},{1, 5},23] (* _Ray Chandler_, Aug 03 2015 *)
%o A085350 (Magma) [2*4^n-3^n: n in [0..30]]; // _Vincenzo Librandi_, Aug 13 2011
%Y A085350 a(n-1) = A080643(n)/2 = A081674(n+1) - A081674(n).
%Y A085350 Cf. A000244 (3^n).
%K A085350 easy,nonn
%O A085350 0,2
%A A085350 _Paul Barry_, Jun 24 2003