A186947 a(n) = 4^n - n*2^n.

1, 2, 8, 40, 192, 864, 3712, 15488, 63488, 257536, 1038336, 4171776, 16728064, 67002368, 268206080, 1073250304, 4293918720, 17177640960, 68714758144, 274867945472, 1099490656256, 4398002470912, 17592093769728, 70368551239680, 281474574057472, 1125899067981824

Offset: 0

Author

Paul Barry, Mar 01 2011

Keywords

• nonn
• easy

Comments

Binomial transform of A186948.

Second binomial transform of A186949.

Examples

			G.f. = 1 + 2*x + 8*x^2 + 40*x^3 + 192*x^4 + 864*x^5 + 3712*x^6 + ... - _Michael Somos_, Jul 18 2018
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-20,16).

Crossrefs

Cf. A000325, A002064, A186948, A186949.

Programs

Magma

[4^n - n*2^n: n in [0..30]]; // G. C. Greubel, Aug 14 2018

Mathematica

Table[4^n-n 2^n,{n,0,30}] (* or *) LinearRecurrence[{8,-20,16},{1,2,8},30] (* Harvey P. Dale, Apr 23 2017 *)

PARI

{a(n) = 2^n * (2^n - n)}; /* Michael Somos, Jul 18 2018 */

Formula

G.f.: (1 - 6*x + 12*x^2)/((1 - 2*x)^2*(1 - 4*x)).

a(n) = 4*a(n-1) + 2^n*(n-2), n >= 1. - Vincenzo Librandi, Mar 13 2011

a(n) = 2^n*A000325(n) = 4^n*A002064(-n) for all n in Z. - Michael Somos, Jul 18 2018

From Elmo R. Oliveira, Sep 15 2024: (Start)

E.g.f.: exp(2*x)*(exp(2*x) - 2*x).

a(n) = 8*a(n-1) - 20*a(n-2) + 16*a(n-3) for n > 2. (End)