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%I M3895 #49 Apr 13 2022 13:25:18
%S 5,20,51,105,190,315,490,726,1035,1430,1925,2535,3276,4165,5220,6460,
%T 7905,9576,11495,13685,16170,18975,22126,25650,29575,33930,38745,
%U 44051,49880,56265,63240,70840,79101,88060,97755,108225,119510,131651,144690,158670
%N Coefficient of x^4 in (1-x-x^2)^(-n).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A006504/b006504.txt">Table of n, a(n) for n = 1..1000</a>
%H G. E. Bergum and V. E. Hoggatt, Jr., <a href="/A006503/a006503.pdf">Numerator polynomial coefficient array for the convolved Fibonacci sequence</a>, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
%H G. E. Bergum and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/14-1/bergum.pdf">Numerator polynomial coefficient array for the convolved Fibonacci sequence</a>, Fib. Quart., 14 (1976), 43-48.
%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences </a>, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
%H P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.
%H Pieter Moree, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Moree/moree12.htm">Convoluted Convolved Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F 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
%F a(n) = n*(n+1)*(n+3)*(n+14)/4!. - _Alois P. Heinz_, Jan 21 2017
%p A006504:=-(5-5*z+z**2)/(z-1)**5; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%o (Haskell)
%o a006504 n = n * (42 + n * (59 + n * (18 + n))) `div` 24
%o -- _Reinhard Zumkeller_, Oct 16 2011
%o (PARI) a(n)=7*n/4+59*n^2/24+3*n^3/4+n^4/24 \\ _Charles R Greathouse IV_, Oct 16 2011
%Y Row n=4 of A144064.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
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