%I #40 Apr 03 2024 20:45:24
%S 0,1,3,7,15,31,62,119,218,381,637,1023,1585,2379,3472,4943,6884,9401,
%T 12615,16663,21699,27895,35442,44551,55454,68405,83681,101583,122437,
%U 146595,174436,206367,242824,284273,331211,384167,443703,510415,584934,667927
%N a(n) = n*(94 + 5*n + 25*n^2 - 5*n^3 + n^4)/120.
%C Previous name was: This sequence is the result of the question: If you have a tall building and 5 plates and you need to find the highest story from which a plate thrown does not break, what is the number of stories you can handle given n tries?
%C Number of compositions with at most five parts and sum at most n. - _Beimar Naranjo_, Mar 12 2024
%H G. C. Greubel, <a href="/A057703/b057703.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Boardman, <a href="http://www.jstor.org/stable/3219201">The Egg-Drop Numbers</a>, Mathematics Magazine, 77 (2004), 368-372. [_Parthasarathy Nambi_, Sep 30 2009]
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = n*(94 + 5*n + 25*n^2 - 5*n^3 + n^4)/120.
%F a(n) = Sum_{j=1..5} binomial(n, j). - _Labos Elemer_
%F G.f.: x*(1 - 3*x + 4*x^2 - 2*x^3 + x^4)/(1-x)^6. - _Colin Barker_, Apr 15 2012
%F E.g.f.: x*(120 + 60*x + 20*x^2 + 5*x^3 + x^4)*exp(x)/120. - _G. C. Greubel_, Jun 05 2019
%p seq(sum(binomial(n,k),k=1..5),n=0..38); # _Zerinvary Lajos_, Dec 13 2007
%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 3, 7, 15, 31}, 60] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2012 *)
%o (PARI) vector(40, n, n--; n*(94+5*n+25*n^2-5*n^3+n^4)/120) \\ _G. C. Greubel_, Jun 05 2019
%o (Magma) [n*(94+5*n+25*n^2-5*n^3+n^4)/120: n in [0..40]]; // _G. C. Greubel_, Jun 05 2019
%o (Sage) [n*(94+5*n+25*n^2-5*n^3+n^4)/120 for n in (0..40)] # _G. C. Greubel_, Jun 05 2019
%o (GAP) List([0..40], n-> n*(94+5*n+25*n^2-5*n^3+n^4)/120) # _G. C. Greubel_, Jun 05 2019
%Y Cf. A004006.
%Y Differences form A055795 + 1 = A000127.
%K nonn,easy
%O 0,3
%A _Leonid Broukhis_, Oct 24 2000
%E More terms and formula from _James A. Sellers_, Oct 25 2000
%E Name changed by _G. C. Greubel_, Jun 06 2019