OFFSET
0,2
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).
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
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 *)
PROG
(Magma) [(n+1)^2*2^n: n in [0..35]]; // Vincenzo Librandi, Aug 21 2011
(PARI) a(n)=(n+1)^2*2^n \\ Charles R Greathouse IV, Apr 07 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved