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a(n) = (Product_{i=1..5} n^i+i)/5!.
9

%I #31 Oct 04 2024 10:58:05

%S 1,6,1221,231880,13443885,340203456,4910472385,47565216504,

%T 342540938025,1962871989130,9382270310061,38701449021984,

%U 141297910237237,465502930269300,1404867737405385,3930816255364816,10296122969028753,25448298063869070,59744930256741205

%N a(n) = (Product_{i=1..5} n^i+i)/5!.

%C See A131685 about well-definedness. - _M. F. Hasler_, May 02 2015

%H T. D. Noe, <a href="/A131675/b131675.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_16">Index entries for linear recurrences with constant coefficients</a>, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

%F G.f.: (135*x^14 +86852*x^13 +5864611*x^12 +109724496*x^11 +782427151*x^10 +2468818430*x^9 +3704965659*x^8 +2710222344*x^7 +952834509*x^6 +152249688*x^5 +9878785*x^4 +212504*x^3 +1245*x^2 -10*x +1) / (x -1)^16. - _Colin Barker_, Apr 24 2015

%t Table[x = 5; Product[(n^k) + k, {k, x}]/x!, {n, 0, 17}] (* _Michael De Vlieger_, Apr 24 2015 *)

%t LinearRecurrence[{16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1},{1,6,1221,231880,13443885,340203456,4910472385,47565216504,342540938025,1962871989130,9382270310061,38701449021984,141297910237237,465502930269300,1404867737405385,3930816255364816},20] (* _Harvey P. Dale_, Oct 04 2024 *)

%o (PARI) Vec((135*x^14 +86852*x^13 +5864611*x^12 +109724496*x^11 +782427151*x^10 +2468818430*x^9 +3704965659*x^8 +2710222344*x^7 +952834509*x^6 +152249688*x^5 +9878785*x^4 +212504*x^3 +1245*x^2 -10*x +1) / (x -1)^16 + O(x^100)) \\ _Colin Barker_, Apr 24 2015

%o (Magma) [((n+1)*(n^2+2)*(n^3+3)*(n^4+4)*(n^5+5))/Factorial(5): n in [0..20]]; // _Vincenzo Librandi_, Apr 25 2015

%o (PARI) A131675(n,k=5)=prod(i=1,k,(n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131676 (k=6), ..., A131680 (k=10). - _M. F. Hasler_, May 02 2015

%K nonn,easy

%O 0,2

%A _Alexander R. Povolotsky_, Sep 15 2007

%E Definition made explicit by _M. F. Hasler_, May 02 2015