A080929 Sequence associated with a(n) = 2*a(n-1) + k*(k+2)*a(n-2).

1, 3, 12, 40, 120, 336, 896, 2304, 5760, 14080, 33792, 79872, 186368, 430080, 983040, 2228224, 5013504, 11206656, 24903680, 55050240, 121110528, 265289728, 578813952, 1258291200, 2726297600, 5888802816, 12683575296, 27246198784

Offset

0,2

Comments

The third column of number triangle A080928.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Formula

G.f.: (1-x)*(1-2*x+4*x^2)/(1-2*x)^3.

For n>0, a(n) = (n+1)*(n+2)*2^(n-2). - Ralf Stephan, Jan 16 2004

a(n) = Sum_{k=0..n} Sum_{i=0..n} (k+1)*binomial(n-1,i). - Wesley Ivan Hurt, Sep 20 2017

From Amiram Eldar, Jan 07 2022: (Start)

Sum_{n>=0} 1/a(n) = 7 - 8*log(2).

Sum_{n>=0} (-1)^n/a(n) = 24*log(3/2) - 9. (End)

Maple

[seq (ceil(binomial(n+2, 2)*2^(n-1)), n=0..30)]; # Zerinvary Lajos, Nov 01 2006

Mathematica

CoefficientList[Series[(1-x)(1-2x+4x^2)/(1-2x)^3, {x, 0, 30}], x] (* Michael De Vlieger, Sep 21 2017 *)
Join[{1}, LinearRecurrence[{6, -12, 8}, {3, 12, 40}, 30]] (* G. C. Greubel, Jul 23 2019 *)

Prog

(Magma) [n eq 0 select 1 else (n+1)*(n+2)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Sep 22 2011
(PARI) vector(30, n, n--; if(n==0, 1, 2^(n-1)*binomial(n+2, 2) )) \\ G. C. Greubel, Jul 23 2019
(Sage) [1]+[2^(n-1)*binomial(n+2, 2) for n in (1..30)] # G. C. Greubel, Jul 23 2019
(GAP) Concatenation([1], List([1..30], n-> 2^(n-1)*Binomial(n+2, 2))); # G. C. Greubel, Jul 23 2019

Crossrefs

Essentially the same as A052482.

Cf. A082140, A082141, A082138, A082139, A080951, A080929, A057711.

Sequence in context: A007993 A293366 A327319 * A052482 A061136 A308648

Adjacent sequences: A080926 A080927 A080928 * A080930 A080931 A080932

Keyword

nonn,easy

Author

Paul Barry, Feb 26 2003

Status

approved