A001874 Convolved Fibonacci numbers. (Formerly M4174 N1738)

1, 6, 27, 98, 315, 924, 2534, 6588, 16407, 39430, 91959, 209034, 464723, 1013292, 2171850, 4584620, 9546570, 19635840, 39940460, 80421600, 160437690, 317354740, 622844730, 1213580820, 2348773525, 4517541378, 8638447293, 16428864606, 31086197469, 58539877020

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

Cf. A049310.

Sequence in context: A277283 A160533 A023005 * A009061 A012320 A097553

Adjacent sequences: A001871 A001872 A001873 * A001875 A001876 A001877

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved