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%I #24 Sep 08 2022 08:44:59
%S 2,10,31,76,161,308,546,912,1452,2222,3289,4732,6643,9128,12308,16320,
%T 21318,27474,34979,44044,54901,67804,83030,100880,121680,145782,
%U 173565,205436,241831,283216,330088,382976,442442,509082,583527,666444,758537
%N a(n) = n*(n+1)*(n+2)*(n^2+7*n+32)/120.
%H Vincenzo Librandi, <a href="/A051747/b051747.txt">Table of n, a(n) for n = 1..1000</a>
%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) = binomial(n+4, n-1)+binomial(n+2, n-1).
%F Convolution of triangular numbers with triangular numbers + 1, i.e. [1, 3, 6, 10, 15, 21, ...] with [2, 4, 7, 11, 16, 22, ...].
%F a(1)=2, a(2)=10, a(3)=31, a(4)=76, a(5)=161, a(6)=308, a(n)=6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - _Harvey P. Dale_, Oct 03 2012
%F G.f.: x*(x^2-2*x+2) / (x-1)^6. - _Colin Barker_, Mar 18 2015
%t Table[(1/120)*n*(n + 1)*(n + 2)*(n^2 + 7*n + 32), {n, 60}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 14 2011 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{2,10,31,76,161,308},60] (* _Harvey P. Dale_, Oct 03 2012 *)
%o (PARI) conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w; t(n)=n*(n+1)/2; u=vector(10,i,t(i)); v=vector(10,i,t(i)+1); conv(u,v)
%o (Magma) [n*(n+1)*(n+2)*(n^2+7*n+32)/120: n in [1..40]]; // _Vincenzo Librandi_, Jun 15 2011
%o (PARI) Vec(x*(x^2-2*x+2)/(x-1)^6 + O(x^100)) \\ _Colin Barker_, Mar 18 2015
%Y Cf. A000217, A000389, A005583.
%K easy,nonn
%O 1,1
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999
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