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A001872
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Convolved Fibonacci numbers.
(Formerly M3476 N1413)
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7
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1, 4, 14, 40, 105, 256, 594, 1324, 2860, 6020, 12402, 25088, 49963, 98160, 190570, 366108, 696787, 1315072, 2463300, 4582600, 8472280, 15574520, 28481220, 51833600, 93914325, 169457708, 304597382, 545556512, 973877245, 1733053440, 3075011478
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OFFSET
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0,2
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REFERENCES
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V. E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.
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. Moree, Convoluted convolved Fibonacci numbers
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
T. Mansour, Generalization of some identities involving the Fibonacci numbers
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FORMULA
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G.f.: 1/(1 - x - x^2)^4.
a(n)= (n+5)*(n+3)*(4*(n+1)*F(n+2)+3*(n+2)*F(n+1))/150, F(n)=A000045(n). - Wolfdieter Lang, Apr 12 2000
For n>3, a(n-3) = sum(h+i+j+k=n, F(h)*F(i)*F(j)*F(k)). - Benoit Cloitre, Nov 01 2002
a(n)=F'''(n+3, 1)/6, i.e. 1/6 times the 3rd derivative of the (n+3)th Fibonacci polynomial evaluated at 1. - T. D. Noe, Jan 18 2006
a(n)=(((-I)^n)/3!)*diff(S(n+3,x),x$3)|_{x=I}. Third derivative of Chebyshev S(n+3,x) polynomial evaluated at x=I (imaginary unit) multiplied by ((-I)^(n-3))/3!. See A049310 for the S-polynomials. W. Lang, Apr 04 2007
a(n)=sum(i=ceil(n/2)..n, (i+1)*(i+2)*(i+3)*binomial(i,n-i))/6. [Vladimir Kruchinin,Apr 26 2011]
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EXAMPLE
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sage: taylor( mul(x^2/(1-x-x^2)^2 for i in xrange(0,2)),x,0,33)# solution>> x^4 + 4*x^5 + 14*x^6 + 40*x^7 + 105*x^8 + 256*x^9 + 594*x^10 + 1324*x^11 + 2860*x^12 + 6020*x^13 + 12402*x^14 +......+ 93914325*x^28 + 169457708*x^29 + 304597382*x^30 + 545556512*x^31 + 973877245*x^32 + 1733053440*x^33, etc... [Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2009]
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MAPLE
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a := n-> (Matrix(8, (i, j)-> if (i=j-1) then 1 elif j=1 then [4, -2, -8, 5, 8, -2, -4, -1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=0..29); [Alois P. Heinz, Aug 15 2008]
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MATHEMATICA
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CoefficientList[Series[1/(1 - x - x^2)^4, {x, 0, 100}], x] (* Stefan Steinerberger, Apr 15 2006 *)
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CROSSREFS
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a(n)= A037027(n+3, 3) (Fibonacci convolution triangle).
Sequence in context: A121593 A160527 A023003 * A054443 A072674 A202900
Adjacent sequences: A001869 A001870 A001871 * A001873 A001874 A001875
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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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EXTENSIONS
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More terms from James A. Sellers, Sep 08 2000
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STATUS
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approved
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