%I #23 Sep 08 2022 08:45:30
%S 1,5,165,4675,65325,543456,3155425,14146210,52259625,166192975,
%T 469090061,1201490445,2839166005,6268589250,13060542825,25881747316,
%U 49095506065,89615392425,158091087925,270522770375,450420100221,731644012660,1162094343345,1808433948150
%N a(n) = (n + 1)*(n^2 + 2)*(n^3 + 3)*(n^4 + 4)/4!.
%C Following my conjecture, computations by _Peter J. C. Moses_, mediation by _Clark Kimberling_ and helpful comments from George E. Andrews, it is now known that a(n) = (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^k + k)/k! is an integer-valued sequence if and only if k belongs to {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21}; this is the case for k=4; see generalization in A131685.
%H T. D. Noe, <a href="/A129995/b129995.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F G.f.: (1-6x+165x^2+2970x^3+22480x^4+55969x^5+51511x^6+16490x^7+1595x^8+25x^9)/(1-x)^11. - _Emeric Deutsch_, Aug 26 2007
%F G.f.: -(1 + x*(-6 + x*(165 + x*(2970 + x*(22480 + x*(55969 + x*(51511 + 5*x*(3298 + x*(319 + 5*x))))))))) / (x - 1)^11. - _Peter J. C. Moses_, Aug 29 2007
%p p:=proc(n,i) mul( n^j+j, j=1..i)/i!; end; [seq(p(n,4),n=0..30)];
%p seq((n+1)*(n^2+2)*(n^3+3)*(n^4+4)/factorial(4), n = 0 .. 20) # _Emeric Deutsch_, Aug 26 2007
%t Table[x = 4; Product[(n^k) + k, {k, x}]/x!, {n, 0, 23}] (* _Michael De Vlieger_, Apr 24 2015 *)
%t LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,5,165,4675,65325,543456,3155425,14146210,52259625,166192975,469090061},30] (* _Harvey P. Dale_, Dec 07 2021 *)
%o (PARI) vector(20,n,n--;(n+1)*(n^2+2)*(n^3+3)*(n^4+4)/4!) \\ _Derek Orr_, Apr 25 2015
%o (Magma) [(n^1 + 1)*(n^2 + 2)*(n^3 + 3)*(n^4 + 4)/24: n in [0..30]]; // _Vincenzo Librandi_, Apr 25 2015
%o (PARI) A129995(n)=(n+1)*(n^2+2)*(n^3+3)*(n^4+4)/12 \\ _M. F. Hasler_, May 02 2015
%Y Cf. A000027 (k=1), A064808 (k=2), A131509 (k=3), this sequence (k=4), A131675 (k=5), ..., A131680 (k=10).
%Y See A131685 for a generalization.
%K nonn,easy
%O 0,2
%A _Alexander R. Povolotsky_, Aug 19 2007, Aug 25 2007