A006504 Coefficient of x^4 in (1-x-x^2)^(-n). (Formerly M3895)

5, 20, 51, 105, 190, 315, 490, 726, 1035, 1430, 1925, 2535, 3276, 4165, 5220, 6460, 7905, 9576, 11495, 13685, 16170, 18975, 22126, 25650, 29575, 33930, 38745, 44051, 49880, 56265, 63240, 70840, 79101, 88060, 97755, 108225, 119510, 131651, 144690, 158670, 173635, 189630, 206701

Offset

1,1

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)

G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48.

Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.

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. 10.

Pieter Moree, Convoluted Convolved Fibonacci Numbers, J. Int. Seq. 7 (2004), Article 04.2.2. See also arXiv:math/0311205 [math.CO], 2003.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

The coefficient of x^4 in (1-x-x^2)^(-n) is the coefficient of x^4 in (1 + x + 2x^2 + 3x^3 + 5x^4)^n. Using the multinomial theorem one then finds that a(n) = 7n/4 + 59*n^2/24 + 3*n^3/4 + n^4/24. - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Sep 03 2003

a(n) = n*(n+1)*(n+3)*(n+14)/4!. - Alois P. Heinz, Jan 21 2017

From Amiram Eldar, Oct 20 2025: (Start)

Sum_{n>=1} 1/a(n) = 2959811/10020010.

Sum_{n>=1} (-1)^(n+1)/a(n) = 160*log(2)/143 - 18401249/30060030. (End)

Maple

A006504:=-(5-5*z+z**2)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

A006504[n_] := n*(n+1)*(n+3)*(n+14)/24; Array[A006504, 50] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {5, 20, 51, 105, 190}, 50] (* Paolo Xausa, Mar 25 2026 *)

Prog

(Haskell)
a006504 n = n * (42 + n * (59 + n * (18 + n))) `div` 24
-- Reinhard Zumkeller, Oct 16 2011
(PARI) a(n)=7*n/4+59*n^2/24+3*n^3/4+n^4/24 \\ Charles R Greathouse IV, Oct 16 2011

Crossrefs

Row n=4 of A144064.

Sequence in context: A358632 A062158 A034133 * A007045 A102227 A173034

Adjacent sequences: A006501 A006502 A006503 * A006505 A006506 A006507

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved