A167784 -id:A167784

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A085350

Binomial transform of poly-Bernoulli numbers A027649.

+10
8

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

(list; graph; refs; listen; history; text; internal format)

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).

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jun 24 2003

STATUS

approved

A167936

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The binomial transform of (0 followed by A077917).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

FORMULA

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)

MATHEMATICA

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

PROG

(Magma) 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

(SageMath)

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

(Python)

def A167936(n): return (1<<n)-3**(n>>1) # Chai Wah Wu, Nov 14 2023

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Nov 15 2009

EXTENSIONS

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

STATUS

approved

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