%I #35 Mar 11 2022 04:52:53
%S 1,120,945,3640,9945,22176,43225,76560,126225,196840,293601,422280,
%T 589225,801360,1066185,1391776,1786785,2260440,2822545,3483480,
%U 4254201,5146240,6171705,7343280,8674225,10178376,11870145,13764520,15877065,18223920,20821801,23688000
%N a(n) = (n+1)*(2*n+1)*(3*n+1)*(4*n+1).
%H Harvey P. Dale, <a href="/A011245/b011245.txt">Table of n, a(n) for n = 0..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 a(n) = A033593(-n).
%F G.f.: (1 + 115*x + 355*x^2 + 105*x^3)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
%F a(0)=1, a(1)=120, a(2)=945, a(3)=3640, a(4)=9945, a(n) = 5*a(n-1)- 10*a(n-2)+ 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, Oct 05 2012
%F E.g.f.: (1 + 119*x + 353*x^2 + 194*x^3 + 24*x^4)*exp(x). - _G. C. Greubel_, Mar 04 2020
%F From _Amiram Eldar_, Mar 11 2022: (Start)
%F Sum_{n>=0} 1/a(n) = (4/3 - 3*sqrt(3)/4)*Pi + 12*log(2) - 27*log(3)/4.
%F Sum_{n>=0} (-1)^n/a(n) = (1 + 4*sqrt(2)/3 - 3*sqrt(3)/2)*Pi - 14*log(2)/3 - 8*sqrt(2)*log(sqrt(2)-1)/3. (End)
%p seq( mul(j*n+1, j=1..4), n=0..30); # _G. C. Greubel_, Mar 04 2020
%t Table[Times@@(Range[4]n+1),{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1}, {1,120,945,3640,9945}, 30] (* _Harvey P. Dale_, Oct 05 2012 *)
%o (Magma) [&*[s*n+1: s in [1..4]]: n in [0..25]]; // _Bruno Berselli_, May 23 2011
%o (PARI) a(n)=24*n^4+50*n^3+35*n^2+10*n+1 \\ _Charles R Greathouse IV_, May 23 2011
%o (Sage) [product(j*n+1 for j in (1..4)) for n in (0..30)] # _G. C. Greubel_, Mar 04 2020
%o (GAP) List([0..30], n-> (n+1)*(2*n+1)*(3*n+1)*(4*n+1) ); # _G. C. Greubel_, Mar 04 2020
%Y Cf. A033593.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_