A001873 Convolved Fibonacci numbers.

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

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) = (((-i)^n)/4!)*(d^4/dx^4)S(n+4,x)|A049310%20for%20the%20S-polynomials.%20-%20_Wolfdieter%20Lang">{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

• T. D. Noe, Table of n, a(n) for n = 0..500
• 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.
• 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).

Programs

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 *)

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 */

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

Extensions

More terms from Wolfdieter Lang, Nov 29 2002