OFFSET
0,2
COMMENTS
Binomial transform of [-5, 10, -5, 0, 0, 0, ...].
If Y is a 5-subset of an n-set X then, for n >= 8, a(n-7) is the number of 4-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 5 * A000292(n).
a(n) = 5*binomial(n+2,3). - Milan Janjic, Dec 28 2007
G.f.: 5*x / (1-x)^4. - R. J. Mathar, Apr 04 2012
a(n) = Sum_{i=0..n} (n+i)*(1+i). - Bruno Berselli, Dec 16 2013
E.g.f.: 5*exp(x)*x*(6 + 6*x + x^2)/6. - Stefano Spezia, Oct 09 2023
From Amiram Eldar, Oct 28 2025: (Start)
Sum_{n>=1} 1/a(n) = 3/10.
Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(2)/5 - 3/2. (End)
EXAMPLE
a(2) = 20 = sum of row 3 terms of triangle A134480: (9 + 7 + 4).
a(3) = 50 = (1, 3, 3, 1) dot (1, 4, 11, 4) = (1 + 12 + 33 + 4).
a(2) = 20 = 2*1 + 3*2 + 4*3; a(5) = 5*1 + 6*2 + 7*3 + 8*4 + 9*5 + 10*6. - Bruno Berselli, Dec 16 2013
MATHEMATICA
CoefficientList[Series[5x/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 29 2012 *)
PROG
(Magma) I:=[0, 5, 20, 50, 100]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 29 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Oct 27 2007
EXTENSIONS
a(0) changed to 0 by Andrew Howroyd, Sep 20 2025
STATUS
approved