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A001872 Convolved Fibonacci numbers.
(Formerly M3476 N1413)
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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).

LINKS

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

FORMULA

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]

EXAMPLE

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, Jun 01 2009]

MAPLE

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]

MATHEMATICA

CoefficientList[Series[1/(1 - x - x^2)^4, {x, 0, 100}], x] (* Stefan Steinerberger, Apr 15 2006 *)

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

More terms from James A. Sellers, Sep 08 2000

STATUS

approved

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Last modified April 19 21:10 EDT 2014. Contains 240777 sequences.