A001874 Convolved Fibonacci numbers.

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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

a(n) = (((-i)^n)/5!)*(d^5/dx^5)S(n+5,x)|A049310%20for%20the%20S-polynomials.%20-%20_Wolfdieter%20Lang">{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

Examples

			G.f. = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 924*x^5 + 2534*x^6 + ...
		

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).

Crossrefs

Cf. A049310.

Programs

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 *)

Sage

taylor( mul(x/(1-x-x^2)^2 for i in range(1,4)),x,0,27) # Zerinvary Lajos, Jun 01 2009

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