%I M4234 N1769 #51 Apr 13 2022 13:25:16

%S 6,40,155,456,1128,2472,4950,9240,16302,27456,44473,69680,106080,

%T 157488,228684,325584,455430,627000,850839,1139512,1507880,1973400,

%U 2556450,3280680,4173390,5265936,6594165,8198880,10126336,12428768,15164952,18400800,22209990

%N Eighth column of quadrinomial coefficients.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H T. D. Noe, <a href="/A001919/b001919.txt">Table of n, a(n) for n = 3..1000</a>

%H L. Carlitz et al., <a href="http://dx.doi.org/10.1016/S0021-9800(66)80057-1">Permutations and sequences with repetitions by number of increases</a>, J. Combin. Theory, 1 (1966), 350-374.

%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_08">Index entries for linear recurrences with constant coefficients</a>, signature (8, -28, 56, -70, 56, -28, 8, -1).

%F a(n) = A008287(n, 7) = binomial(n+2, 5)*(n^2+21*n+180 )/42, n >= 3.

%F G.f.: (x^3)*(6-8*x+3*x^2 )/(1-x)^8. Numerator polynomial is N4(7, x) from array A063421.

%F a(n) = n(n^2-1)(n^2-4)(n^2+21n+180)/5040. - _Emeric Deutsch_, Jan 27 2005

%F a(n) = 6*C(n,3) + 16*C(n,4) + 15*C(n,5) + 6*C(n,6) + C(n,7) (see comment in A071675). - _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 22 2012

%F a(3)=6, a(4)=40, a(5)=155, a(6)=456, a(7)=1128, a(8)=2472, a(9)=4950, a(10)=9240, a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)- 28*a(n-6)+ 8*a(n-7)-a(n-8). - _Harvey P. Dale_, Mar 27 2013

%p seq(n*(n^2-1)*(n^2-4)*(n^2+21*n+180)/5040,n=3..34); # _Emeric Deutsch_, Jan 27 2005

%p A001919:=(3*z**2-8*z+6)/(z-1)**8; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t Table[n*(n^2 - 1)*(n^2 - 4)*(n^2 + 21*n + 180)/5040, {n, 3, 50}] (* _T. D. Noe_, Aug 17 2012 *)

%t LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{6,40,155,456,1128,2472,4950,9240},40] (* _Harvey P. Dale_, Mar 27 2013 *)

%K nonn,easy

%O 3,1

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, Jan 27 2005