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%I M4729 N2021 #54 Jun 24 2023 21:41:23
%S 10,80,365,1246,3535,8800,19855,41470,81367,151580,270270,464100,
%T 771290,1245488,1960610,3016820,4547840,6729800,9791859,14028850,
%U 19816225,27627600,38055225,51833730,69867525,93262260,123360780,161784040,210477476,271763360
%N Tenth column of quintinomial coefficients.
%C In the Carlitz et al. reference a(n)= Q_{5,n+2}(2), n >= 0, with a(n)=binomial(11+n,n+2)-(n+3)*binomial(n+6,n+2), (eq.(3.3), p. 356, with n=5, m->n+2,r=2). Q_{5,m}(2) is the number of sequences (i_1,i_2,...,i_m) with i_s, s=1,...,m, from {1,2,3,4,5} (repetitions allowed), with exactly 2 increases between successive elements (first position is counted as an increase).
%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 Vincenzo Librandi, <a href="/A000575/b000575.txt">Table of n, a(n) for n = 0..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, p. 351.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).
%F a(n) = A035343(n+3, 9) = binomial(n+6, 6)*(n^3+42*n^2+677*n+5040)/(9!/6!).
%F G.f.: (10-20*x+15*x^2-4*x^3)/(1-x)^10; numerator polynomial is N5(9, x) from the array A063422.
%F a(n) = 10*C(n+3,3) + 40*C(n+3,4) + 65*C(n+3,5) + 56*C(n+3,6) + 28*C(n+3,7) + 8*C(n+3,8) + C(n+3,9) (see comment in A213887). - _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 22 2012
%F a(n) = Sum_{k=1..10} (-1)^k * binomial(10,k) * a(n-k), a(0)=10. - _G. C. Greubel_, Aug 03 2015
%F a(n) = [x^9] (1+x+x^2+x^3+x^4)^(n+3). - _Joerg Arndt_, Aug 04 2015
%t CoefficientList[Series[(10-20*x+15*x^2-4*x^3)/(1-x)^10,{x,0,50}],x](* _Vincenzo Librandi_, Mar 28 2012 *)
%o (PARI) a(n) = polcoeff((1+x+x^2+x^3+x^4)^(n+3), 9); \\ _Joerg Arndt_, Aug 04 2015
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E Comments and more terms from _Wolfdieter Lang_, Aug 29 2001
%E More terms from _Sean A. Irvine_, Nov 24 2010
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