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%I M4702 #44 Jun 28 2023 20:42:27
%S 1,10,44,135,336,728,1428,2598,4455,7282,11440,17381,25662,36960,
%T 52088,72012,97869,130986,172900,225379,290444,370392,467820,585650,
%U 727155,895986,1096200,1332289,1609210,1932416,2307888,2742168,3242393,3816330,4472412,5219775
%N Quadrinomial coefficients.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. K. Guy, <a href="/A005712/a005712.pdf">Letter to N. J. A. Sloane, 1987</a>
%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_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n)= binomial(n+1, 3)*(n^3+15*n^2+86*n-120)/120, n >= 2.
%F G.f.: (x^2)*(1+3*x-5*x^2+2*x^3)/(1-x)^7. (numerator polynomial is N4(6, x) from A063421).
%F a(0)=1, a(1)=10, a(2)=44, a(3)=135, a(4)=336, a(5)=728, a(6)=1428, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - _Harvey P. Dale_, Jun 23 2011
%F a(n) = binomial(n,2) + 7*binomial(n,3) + 10*binomial(n,4) + 5*binomial(n,5) + binomial(n,6) (see comment in A071675). - _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 22 2012
%p A005720:=-(1+3*z-5*z**2+2*z**3)/(z-1)**7; [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,44,135,336,728,1428},40] (* or *) Table[Binomial[n+1,3] (n^3+15n^2+86n-120)/120,{n,2,41}] (* _Harvey P. Dale_, Jun 23 2011 *)
%o (PARI) a(n)=(n^6 + 15*n^5 + 85*n^4 - 135*n^3 - 86*n^2 + 120*n)/720 \\ _Charles R Greathouse IV_, Jun 23 2011
%Y a(n)= A008287(n, 6), n >= 2 (seventh column of quadrinomial coefficients).
%K nonn,easy
%O 2,2
%A _N. J. A. Sloane_
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