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n + n(n+1) + n(n+1)(n+2) + ..., with n terms.
2

%I #17 Aug 19 2024 12:16:03

%S 1,8,75,984,17045,366288,9374967,278095760,9375293169,353906211240,

%T 14785127222723,677150215857192,33734100501544365,1816008001717251104,

%U 105048613959883117935,6497985798745934394528,427999600108502895779657

%N n + n(n+1) + n(n+1)(n+2) + ..., with n terms.

%H Robert Israel, <a href="/A111685/b111685.txt">Table of n, a(n) for n = 1..364</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RisingFactorial.html">Rising Factorial.</a>

%e a(3) = 3+ 3*4 +3*4*5 = 75.

%p seq(add(pochhammer(n,i),i=1..n),n=1..20); # _Robert Israel_, Aug 18 2024

%t Table[Total[Rest[FoldList[Times,1,Range[n,2n-1]]]],{n,20}] (* _Harvey P. Dale_, Apr 22 2011 *)

%t Table[Total[Pochhammer[n, Range[n]]], {n, 20}] (* _T. D. Noe_, Apr 22 2011 *)

%o (PARI) a(n) = sum(k=0,n-1,prod(j=0,k,n+j)) \\ _Rick L. Shepherd_, Aug 16 2005

%Y Cf. A054654.

%K nonn

%O 1,2

%A _Amarnath Murthy_, Aug 16 2005

%E More terms from _Rick L. Shepherd_, Aug 16 2005