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A001872 Convolved Fibonacci numbers.
(Formerly M3476 N1413)
12
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
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
D. Birmajer, J. Gil and M. Weiner, Linear recurrence sequences and their convolutions via Bell polynomials, arXiv:1405.7727 [math.CO], 2014.
D. Birmajer, J. B. Gil, M. D. Weiner, Linear Recurrence Sequences and Their Convolutions via Bell Polynomials, Journal of Integer Sequences, 18 (2015), #15.1.2.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
V. E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.
M. Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8, section 3.
T. Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
FORMULA
G.f.: 1/(1 - x - x^2)^4.
a(n) = A037027(n+3, 3) (Fibonacci convolution triangle).
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!)*(d^3/dx^3)S(n+3,x)|_{x=i}, where i is the imaginary unit. Third derivative of Chebyshev S(n+3,x) polynomial evaluated at x=i multiplied by ((-i)^(n-3))/3!. See A049310 for the S-polynomials. - Wolfdieter Lang, Apr 04 2007
a(n) = Sum_{i=ceiling(n/2)..n} (i+1)*(i+2)*(i+3)*binomial(i,n-i)/6. - Vladimir Kruchinin, Apr 26 2011
Recurrence: a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 5*a(n-4) + 8*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8). - Fung Lam, May 11 2014
n*a(n) - (n+3)*a(n-1) - (n+6)*a(n-2) = 0, n > 1. - Michael D. Weiner, Nov 18 2014
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 *)
PROG
(PARI) Vec( 1/(1 - x - x^2)^4 + O(x^66) ) \\ Joerg Arndt, May 12 2014
(Magma) [(n+5)*(n+3)*(4*(n+1)*Fibonacci(n+2)+3*(n+2)*Fibonacci(n+1))/150: n in [0..30]]; // Vincenzo Librandi, Nov 19 2014
CROSSREFS
Sequence in context: A121593 A160527 A023003 * A054443 A281766 A072674
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)