Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #31 Oct 09 2024 07:35:36
%S 0,24,84,204,414,750,1254,1974,2964,4284,6000,8184,10914,14274,18354,
%T 23250,29064,35904,43884,53124,63750,75894,89694,105294,122844,142500,
%U 164424,188784,215754,245514,278250,314154,353424,396264,442884,493500,548334
%N a(n) = (n+6)*(n+1)*(n^2 + 7*n + 16)/4.
%C In general, a sequence of the form a(n) = Sum_{k=1..n} (k+x+3)!/(k+x)! will have a closed form of n/4 * (n + 5 + 2*x)*(n^2 + 5*n + 2*x*n + 10 + 2*x^2 + 10*x). - _Gary Detlefs_, Aug 10 2010
%H G. C. Greubel, <a href="/A168538/b168538.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F From _R. J. Mathar_, Mar 21 2010: (Start)
%F a(n) = 6*A063258(n).
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F G.f.: 6*(x-2)*(x^2 - 2*x + 2)/(x-1)^5. (End)
%F From _Gary Detlefs_, Aug 10 2010: (Start)
%F a(n) = Sum_{k=1..n} (k+3)*(k+2)*(k+1), with offset 0.
%F a(n) = (n/4)*(n+5)*(n^2 + 5*n + 10), with offset 0. (End)
%F E.g.f.: (1/4)*(96 + 240*x + 120*x^2 + 20*x^3 + x^4)*exp(x). - _G. C. Greubel_, Jul 25 2016
%t LinearRecurrence[{5,-10,10,-5,1},{0, 24, 84, 204, 414}, 25] (* _G. C. Greubel_, Jul 25 2016 *)
%t Table[1/4 (n+6)(n+1)(n^2+7n+16),{n,-1,40}] (* _Harvey P. Dale_, Jul 07 2019 *)
%o (PARI) a(n) = (n+6)*(n+1)*(n^2 + 7*n + 16)/4; \\ _Michel Marcus_, Jan 10 2015
%o (Magma) [(n+6)*(n+1)*(n^2+7*n+16)/4: n in [-1..35]]; // _Vincenzo Librandi_, Jul 26 2016
%Y Cf. A063258.
%K nonn,easy
%O -1,2
%A Kristo Jorgenson (kristoj(AT)me.com), Nov 29 2009
%E Edited by _N. J. A. Sloane_, Nov 29 2009