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A001874 Convolved Fibonacci numbers.
(Formerly M4174 N1738)
4
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 (list; graph; refs; listen; history; text; internal format)
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) = 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

Conjecture: satisfies a linear recurrence having signature (6, -9, -10, 30, 6, -41, -6, 30, 10, -9, -6, -1). - Harvey P. Dale, Sep 09 2019

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.

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) then 1 elif j=1 then [6, -9, -10, 30, 6, -41, -6, 30, 10, -9, -6, -1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=0..24); # 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 xrange(1, 4)), x, 0, 27) # Zerinvary Lajos, Jun 01 2009

CROSSREFS

Sequence in context: A277283 A160533 A023005 * A009061 A012320 A097553

Adjacent sequences:  A001871 A001872 A001873 * A001875 A001876 A001877

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified November 21 09:14 EST 2019. Contains 329362 sequences. (Running on oeis4.)