OFFSET
0,2
COMMENTS
a(n) = (((-i)^n)/5!)*(d^5/dx^5)S(n+5,x)|_{x=i}, where i is the imaginary unit. Fifth derivative of Chebyshev S(n+5,x) polynomials evaluated at x=i multiplied by ((-i)^n)/5!. See A049310 for the S-polynomials. - Wolfdieter Lang, Apr 04 2007
a(n) is the number of weak compositions of n in which exactly 5 parts are 0 and all other parts are either 1 or 2. - Milan Janjic, Jun 28 2010
REFERENCES
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..500
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Index entries for linear recurrences with constant coefficients, signature (6,-9,-10,30,6,-41,-6,30,10,-9,-6,-1).
FORMULA
G.f.: ( 1 - x - x^2 )^(-6).
a(n) = F'''''(n+5, 1)/5!, i.e., 1/5! times the 5th derivative of the (n+5)th Fibonacci polynomial evaluated at 1. - T. D. Noe, Jan 18 2006
EXAMPLE
G.f. = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 924*x^5 + 2534*x^6 + ...
MAPLE
a:= n-> (Matrix(12, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [6, -9, -10,
30, 6, -41, -6, 30, 10, -9, -6, -1][i], 0)))^n)[1, 1]:
seq(a(n), n=0..31); # Alois P. Heinz, Aug 15 2008
MATHEMATICA
nn = 30; t = CoefficientList[Series[1/(1 - x - x^2)^6, {x, 0, nn}], x] (* T. D. Noe, Aug 10 2012 *)
PROG
(Sage) taylor( mul(x/(1-x-x^2)^2 for i in range(1, 4)), x, 0, 27) # Zerinvary Lajos, Jun 01 2009
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved