A167784 -id:A167784 - cp's OEIS frontend

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

Paul Barry, Jun 24 2003

Binomial transform is A085351.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -12).

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

Cf. A000244 (3^n).

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

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 *)

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)

A167936 a(n) = 2^n - A108411(n).

Original entry on oeis.org

0, 1, 1, 5, 7, 23, 37, 101, 175, 431, 781, 1805, 3367, 7463, 14197, 30581, 58975, 124511, 242461, 504605, 989527, 2038103, 4017157, 8211461, 16245775, 33022991, 65514541, 132623405, 263652487, 532087943, 1059392917, 2133134741, 4251920575, 8546887871

Offset: 0

Paul Curtz, Nov 15 2009

The binomial transform of (0 followed by A077917).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

Cf. A005061, A077917, A083658, A085350, A108411, A167762, A167784.

I:=[0,1,1]; [n le 3 select I[n] else 2*Self(n-1) +3*Self(n-2) -6*Self(n-3): n in [1..40]]; // G. C. Greubel, Sep 10 2023

LinearRecurrence[{2,3,-6}, {0,1,1}, 50] (* G. C. Greubel, Jul 01 2016 *)

def A167936(n): return (1<>1) # Chai Wah Wu, Nov 14 2023

def A167936(n): return 2^n - ((n+1)%2)*3^(n//2) - (n%2)*3^((n-1)//2)
[A167936(n) for n in range(41)] # G. C. Greubel, Sep 10 2023

a(n) = A167762(n+1) - A167762(n).
a(n+1) - a(n) = A167784(n).
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3).
G.f.: x*(1-x)/((1-2*x)*(1-3*x^2)).
a(2n) = A005061(n), a(2n+1) = A085350(n).
a(n) - 2*a(n-1) = (-1)^(n+1)*A083658(n+1).
From G. C. Greubel, Sep 10 2023: (Start)
a(n) = (1/2)*(2^(n+1) - (1+(-1)^n)*3^(n/2) - (1-(-1)^n)*3^((n-1)/2)).
E.g.f.: exp(2*x) - cosh(sqrt(3)*x) - (1/sqrt(3))*sinh(sqrt(3)*x). (End)

Edited and extended by R. J. Mathar, Feb 27 2010

cp's OEIS Frontend

A085350 Binomial transform of poly-Bernoulli numbers A027649.

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