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A001873
Convolved Fibonacci numbers.
13
1, 5, 20, 65, 190, 511, 1295, 3130, 7285, 16435, 36122, 77645, 163730, 339535, 693835, 1399478, 2790100, 5504650, 10758050, 20845300, 40075630, 76495450, 145052300, 273381350, 512347975, 955187033, 1772132390, 3272875935, 6018885570, 11024814945, 20118711993
OFFSET
0,2
COMMENTS
a(n) = (((-i)^n)/4!)*(d^4/dx^4)S(n+4,x)|_{x=i}, where i is the imaginary unit. Fourth derivative of Chebyshev S(n+4,x) polynomial evaluated at x=i multiplied by ((-i)^n)/4!. See A049310 for the S-polynomials. - Wolfdieter Lang, Apr 04 2007
a(n) = number of weak compositions of n in which exactly 4 parts are 0 and all other parts are either 1 or 2. - Milan Janjic, Jun 28 2010
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
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Verner E. Hoggatt, Jr. and Marjorie Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.
Vitaly M. Khamitov, Dmitriy Dmitrishin, Alexander Stokolos, and Daniel Gray, Convolved Numbers of k-sections of the Fibonacci Sequence: Properties, Consequences, arXiv:2603.08636 [math.CA], 2026. See p. 5.
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 18.
Paul R. Stein and Michael S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
Paul R. Stein and Michael S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers [Corrected annotated scanned copy]
Michael S. Waterman, Home Page (contains copies of his papers)
Index entries for linear recurrences with constant coefficients, signature (5,-5,-10,15,11,-15,-10,5,5,1).
FORMULA
G.f.: 1/(1-x-x^2)^5.
From Wolfdieter Lang, Nov 29 2002: (Start)
a(n) = Sum_{m=0.. floor(n/2)} binomial(4+n-m, 4)*binomial(n-m, m).
a(n) = ((1368 + 970*n + 215*n^2 + 15*n^3)*(n+1)*F(n+2) + 2*(408 + 305*n + 70*n^2 + 5*n^3)*(n+2)*F(n+1))/(4!*5^3), with F(n) = A000045(n). (End)
a(n) = F''''(n+4, 1)/24, i.e., 1/24 times the 4th derivative of the (n+4)th Fibonacci polynomial evaluated at 1. - T. D. Noe, Jan 18 2006
Recurrence: a(n) = 5*a(n-1) - 5*a(n-2) - 10*a(n-3) + 15*a(n-4) + 11*a(n-5) - 15*a(n-6) - 10*a(n-7) + 5*a(n-8) + 5*a(n-9) + a(n-10). - Fung Lam, May 11 2014
For n > 1, a(n) = (4/n+1)*a(n-1)+(8/n+1)*a(n-2). - Tani Akinari, Sep 14 2023
MAPLE
a:= n-> (Matrix(10, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [5, -5, -10, 15, 11, -15, -10, 5, 5, 1][i], 0 )))^n)[1, 1]: seq(a(n), n=0..40); # Alois P. Heinz, Aug 15 2008
MATHEMATICA
nn = 30; CoefficientList[Series[1/(1 - x - x^2)^5, {x, 0, nn}], x] (* T. D. Noe, Aug 10 2012 *)
LinearRecurrence[{5, -5, -10, 15, 11, -15, -10, 5, 5, 1}, {1, 5, 20, 65, 190, 511, 1295, 3130, 7285, 16435}, 40] (* Harvey P. Dale, Aug 10 2021 *)
PROG
(Maxima) a[n]:=if n<2 then 4*n+1 else (4/n+1)*a[n-1]+(8/n+1)*a[n-2];
makelist(a[n], n, 0, 50); /* Tani Akinari, Sep 14 2023 */
CROSSREFS
Sequence in context: A277212 A160528 A023004 * A391791 A120297 A271066
KEYWORD
nonn,easy
EXTENSIONS
More terms from Wolfdieter Lang, Nov 29 2002
STATUS
approved