A001873 Convolved Fibonacci numbers. (Formerly M3899 N1600)

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

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

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.

P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers [Corrected annotated scanned copy]

M. 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 * A120297 A271066 A271599

Adjacent sequences: A001870 A001871 A001872 * A001874 A001875 A001876

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Wolfdieter Lang, Nov 29 2002

Status

approved