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A210569 a(n) = (n-3)*(n-2)*(n-1)*n*(n+1)/30.

%I A210569 #59 Jun 02 2022 14:50:27
%S A210569 0,0,0,0,4,24,84,224,504,1008,1848,3168,5148,8008,12012,17472,24752,
%T A210569 34272,46512,62016,81396,105336,134596,170016,212520,263120,322920,
%U A210569 393120,475020,570024,679644,805504,949344,1113024,1298528,1507968,1743588,2007768,2303028
%N A210569 a(n) = (n-3)*(n-2)*(n-1)*n*(n+1)/30.
%C A210569 The following sequences are provided by the formula n*binomial(n,k) - binomial(n,k+1) = k*binomial(n+1,k+1):
%C A210569 . A000217(n)   for k=1,
%C A210569 . A007290(n+1) for k=2,
%C A210569 . A050534(n)   for k=3,
%C A210569 . a(n)         for k=4,
%C A210569 . A000910(n+1) for k=5.
%C A210569 Sum of reciprocals of a(n), for n>3: 5/16.
%C A210569 From a(2), convolution of oblong numbers (A002378) with themselves. - _Bruno Berselli_, Oct 24 2016
%H A210569 Bruno Berselli, <a href="/A210569/b210569.txt">Table of n, a(n) for n = 0..1000</a>
%H A210569 C. P. Neuman and D. I. Schonbach, <a href="http://dx.doi.org/10.1137/1019006">Evaluation of sums of convolved powers using Bernoulli numbers</a>, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 3. - _N. J. A. Sloane_, Mar 23 2014
%H A210569 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A210569 G.f.: 4*x^4/(1-x)^6.
%F A210569 a(n) = n*binomial(n,4)-binomial(n,5) = 4*binomial(n+1,5) = 4*A000389(n+1).
%F A210569 a(n) = 2*A177747(2*n-7), with A177747(-7) = A177747(-5) = A177747(-3) = A177747(-1) = 0.
%F A210569 (n-4)*a(n) = (n+1)*a(n-1).
%F A210569 E.g.f.: (1/30)*x^4*(5+x)*exp(x). - _G. C. Greubel_, May 23 2022
%F A210569 Sum_{n>=4} (-1)^n/a(n) = 20*log(2) - 655/48. - _Amiram Eldar_, Jun 02 2022
%p A210569 f:=n->(n^5-5*n^3+4*n)/30;
%p A210569 [seq(f(n),n=0..50)]; # _N. J. A. Sloane_, Mar 23 2014
%t A210569 LinearRecurrence[{6,-15,20,-15,6,-1}, {0,0,0,0,4,24}, 39]
%t A210569 CoefficientList[Series[4x^4/(1-x)^6, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)
%t A210569 Times@@@Partition[Range[-3,40],5,1]/30 (* _Harvey P. Dale_, Sep 19 2020 *)
%o A210569 (Magma) [4*Binomial(n+1,5): n in [0..38]];
%o A210569 (Maxima) makelist(coeff(taylor(4*x^4/(1-x)^6, x, 0, n), x, n), n, 0, 38);
%o A210569 (PARI) a(n)=(n-3)*(n-2)*(n-1)*n*(n+1)/30 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A210569 (SageMath) [4*binomial(n+1,5) for n in (0..40)] # _G. C. Greubel_, May 23 2022
%Y A210569 Cf. A000217, A000389, A000910, A002378, A007290, A050534, A177747.
%Y A210569 First differences are in A033488.
%K A210569 nonn,easy
%O A210569 0,5
%A A210569 _Bruno Berselli_, Mar 23 2012