OFFSET
0,2
COMMENTS
With three leading zeros, 3rd binomial transform of (0,0,0,1,0,0,0,0,...). - Paul Barry, Mar 07 2003
Number of n-permutations (n=4) of 4 objects u, v, w, z, with repetition allowed, containing exactly three u's. - Zerinvary Lajos, May 23 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Idempotent Number.
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).
FORMULA
a(n) = 3^n*binomial(n+3, 3).
a(n) = A027465(n+4, 4).
G.f.: 1/(1 - 3*x)^4.
With three leading zeros, a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 81*a(n-4), a(0) = a(1) = a(2) = 0, a(3) = 1. - Paul Barry, Mar 07 2003
With three leading zeros, C(n, 3)*3^(n-3) is the second binomial transform of C(n, 3). - Paul Barry, Jul 24 2003
E.g.f.: (1/2)*(2 + 18*x + 27*x^2 + 9*x^3)*exp(3*x). - Franck Maminirina Ramaharo, Nov 23 2018
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 36*log(3/2) - 27/2.
Sum_{n>=0} (-1)^n/a(n) = 144*log(4/3) - 81/2. (End)
MAPLE
seq(3^n*binomial(n+3, 3), n=0..30); # Zerinvary Lajos, Dec 21 2006
MATHEMATICA
CoefficientList[Series[1/(1-3x)^4, {x, 0, 30}], x] (* or *) LinearRecurrence[ {12, -54, 108, -81}, {1, 12, 90, 540}, 30] (* Harvey P. Dale, Jul 27 2017 *)
PROG
(SageMath) [3^n*binomial(n+3, 3) for n in range(30)] # Zerinvary Lajos, Mar 10 2009
(Magma) [3^n* Binomial(n+3, 3): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011
(PARI) a(n) = 3^n*binomial(n+3, 3) \\ Charles R Greathouse IV, Oct 03 2016
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved