A215149 a(n) = n * (1 + 2^(n-1)).

0, 2, 6, 15, 36, 85, 198, 455, 1032, 2313, 5130, 11275, 24588, 53261, 114702, 245775, 524304, 1114129, 2359314, 4980755, 10485780, 22020117, 46137366, 96469015, 201326616, 419430425, 872415258, 1811939355, 3758096412, 7784628253, 16106127390, 33285996575, 68719476768, 141733920801, 292057776162

Offset: 0

Author

Paul Curtz, Aug 04 2012

Keywords

• nonn
• easy

Comments

Related to Bernoulli numbers.

Essentially the same as A135854.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Crossrefs

Cf. A001477, A001787, A001792, A027641, A027642, A094373, A132045, A135854, A157809, A164555, A164558.

Programs

Magma

[n*(1 + 2^(n-1)): n in [0..40]]; // G. C. Greubel, Apr 19 2018

Mathematica

Table[n(1+2^(n-1)),{n,0,40}] (* or *) LinearRecurrence[{6,-13,12,-4},{0,2,6,15}, 40] (* Harvey P. Dale, Oct 18 2013 *)

PARI

a(n) = n*(1+2^(n-1)) \\ Michel Marcus, Mar 10 2013

Python

def A215149(n): return n*(pow(2,n)+2)//2
print([A215149(n) for n in range(41)]) # G. C. Greubel, Jan 18 2025

Formula

a(n) = (A157809(n) - A164555(n)) / A027642(n).

a(n) = n (the nonnegative integers A001477(n)) + n*2^(n-1) (their binomial transform A001787(n)).

a(n+1) - a(n) = 2,4,9,21,... = A001792(n) + 1.

a(n+1) - 2*a(n) = 2 before A132045(n+1).

a(n) is the binomial transform of b(n) = 0,2,2,3,4,5,... = A001477(n) with 2 instead of 1. b(n) = (A164558(n) - A027641(n))/A027642(n)?

G.f.: x*(2-6*x+5*x^2) / ( (1-x)^2*(1-2*x)^2 ). - R. J. Mathar, Aug 06 2012

E.g.f.: x*exp(x)*(1 + exp(x)). - G. C. Greubel, Jan 18 2025

a(n) = n * A094373(n). - Alois P. Heinz, Jan 18 2025