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A001872
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Convolved Fibonacci numbers.
(Formerly M3476 N1413)
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12
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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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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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FORMULA
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G.f.: 1/(1 - x - x^2)^4.
a(n) = A037027(n+3, 3) (Fibonacci convolution triangle).
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
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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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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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