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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.
• P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
• Pieter Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
• 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).

Crossrefs

Row n=4 of A144064.

Programs

Haskell

a006504 n = n * (42 + n * (59 + n * (18 + n))) `div` 24
-- Reinhard Zumkeller, Oct 16 2011

Maple

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

PARI

a(n)=7*n/4+59*n^2/24+3*n^3/4+n^4/24 \\ Charles R Greathouse IV, Oct 16 2011

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