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A001874
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Convolved Fibonacci numbers.
(Formerly M4174 N1738)
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5
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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G.f. = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 924*x^5 + 2534*x^6 + ...
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MAPLE
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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
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MATHEMATICA
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nn = 30; t = CoefficientList[Series[1/(1 - x - x^2)^6, {x, 0, nn}], x] (* T. D. Noe, Aug 10 2012 *)
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PROG
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(Sage) taylor( mul(x/(1-x-x^2)^2 for i in range(1, 4)), x, 0, 27) # Zerinvary Lajos, Jun 01 2009
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CROSSREFS
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Cf. A049310.
Sequence in context: A277283 A160533 A023005 * A009061 A012320 A097553
Adjacent sequences: A001871 A001872 A001873 * A001875 A001876 A001877
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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STATUS
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approved
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