A014477 Expansion of (1 + 2*x)/(1 - 2*x)^3.

1, 8, 36, 128, 400, 1152, 3136, 8192, 20736, 51200, 123904, 294912, 692224, 1605632, 3686400, 8388608, 18939904, 42467328, 94633984, 209715200, 462422016, 1015021568, 2218786816, 4831838208, 10485760000, 22682796032, 48922361856, 105226698752, 225754218496

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The sequence 0,1,8,... has a(n) = n^2*2^(n-1) and is the binomial transform of the hexagonal numbers A000384 (with leading 0). - Paul Barry, Jun 09 2003

As 0,1,8,... this is n^2*2^(n-1), the binomial transform of the hexagonal numbers A000384 (include the leading 0). Partial sums are A036826. - Paul Barry, Jun 10 2003

Sequence gives total value of all possible sums of distinct odd integers with maximum term less than 2n+1. E.g., for a(3) we can have the sums 1, 3, 5, 1+3, 1+5, 3+5, 1+3+5, which sum to 1+3+5+4+6+8+9 = 36. - Jon Perry, Feb 06 2004

Number of edges on a partially truncated (n+1)-cube (column 2 of A271316).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Crossrefs

Cf. A007758, A036826, A118414, A355234.

Programs

Magma

[(n+1)^2*2^n: n in [0..35]]; // Vincenzo Librandi, Aug 21 2011

Maple

a:=n->sum(binomial(n,j)*n*j,j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Oct 19 2006
a:=n->sum(n*numbcomb(n)/2, j=1..n): seq(a(n), n=1..25); # Zerinvary Lajos, Apr 25 2007

Mathematica

f[n_]:=(n^2*2^n)/2;Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2009 *)

PARI

a(n)=(n+1)^2*2^n \\ Charles R Greathouse IV, Apr 07 2016

Formula

O.g.f.: (1 + 2*x)/(1 - 2*x)^3 (see the name).

a(n) = (n+1)^2*2^n = A007758(n+1)/2. - Henry Bottomley, Jun 13 2001

The binomial transform of 0, 1, 8, ... is A077616. - Paul Barry, Jul 24 2003

a(1)=1, a(n) = 2a(n-1) + (2n-1)*2^(n-1). - Jon Perry, Feb 06 2004

a(n) = sum of (n+1)-th row of the triangle in A118416. - Reinhard Zumkeller, Apr 27 2006

a(n) = Sum_{j=0..n} binomial(n,j)*n*j. - Zerinvary Lajos, Oct 19 2006

E.g.f.: exp(2*x)*(1 + 6*x + 8*x^2/2!). - Wolfdieter Lang, Jul 29 2017

Sum_{n>=0} 1/a(n) = Pi^2/6 - log(2)^2. - Daniel Suteu, Oct 31 2017

Sum_{n>=0} (-1)^n/a(n) = -2 * Li_2(-1/2) = -2 * A355234. - Amiram Eldar, Oct 01 2022