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%I #39 Sep 08 2022 08:45:31
%S 1,4,33,220,1005,3456,9709,23528,50985,101260,187561,328164,547573,
%T 877800,1359765,2044816,2996369,4291668,6023665,8303020,11260221,
%U 15047824,19842813,25849080,33300025,42461276,53633529,67155508,83407045,102812280,125842981
%N a(n) = (n + 1)*(n^2 + 2)*(n^3 + 3)/6.
%C See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0. For k=3, A131685(k)=1, which implies that this is a well defined integer sequence. - _Alexander R. Povolotsky_, May 18 2015
%H M. F. Hasler, <a href="/A131509/b131509.txt">Table of n, a(n) for n = 0..100</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F G.f.: (1 -3x +26x^2 +38x^3 +53x^4 +5x^5)/(1-x)^7. - _Emeric Deutsch_, Aug 23 2007
%p p:=proc(n,i) mul( n^j+j, j=1..i)/i!; end; [seq(p(n,3),n=0..30)];
%p seq((1/6)*(n+1)*(n^2+2)*(n^3+3),n=0..25); # _Emeric Deutsch_, Aug 23 2007
%t Table[x = 3; Product[(n^k) + k, {k, x}]/6, {n, 0, 27}] (* _Michael De Vlieger_, Apr 24 2015 *)
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,4,33,220,1005,3456,9709},40] (* _Harvey P. Dale_, Oct 18 2016 *)
%o (Maxima) A131509(n):=(n^1 + 1)*(n^2 + 2)*(n^3 + 3)/6$
%o makelist(A131509(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */
%o (PARI) vector(20,n,n--;(n+1)*(n^2+2)*(n^3+3)/3!) \\ _Derek Orr_, Apr 25 2015
%o (Magma) [(n^1 + 1)*(n^2 + 2)*(n^3 + 3)/6: n in [0..30]]; // _Vincenzo Librandi_, Apr 25 2015
%o (PARI) A131509(n)=(n+1)*(n^2+2)*(n^3+3)/6 \\ _M. F. Hasler_, May 02 2015
%Y Cf. A000027 (k=1), A064808 (k=2), this sequence (k=3), A129995 (k=4), A131675 (k=5), ..., A131680 (k=10).
%K nonn,easy
%O 0,2
%A _Alexander R. Povolotsky_, Aug 13 2007, Aug 25 2007
%E Corrected and extended by _R. J. Mathar_ and _Emeric Deutsch_, Aug 21 2007
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