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%I #2 Mar 31 2012 13:20:26
%S 1,9,34,265,186,1141,2868,31401,18635,477301,91192,8051069,4508441,
%T 3336145,22048024,410111791,223063947,3057889621,823596665,706952715,
%U 125961187,6173866701,9838037952,521135614075,275363139571
%N Numerator of (1^2/n + 2^2/(n-1) + ... + k^2/(n-k+1) + ... + (n-1)^2/2 + n^2/1).
%C p divides a(p-1) for prime p>2. p divides a(2p-1) for all prime p. p divides a(3p-1) for all prime p. p divides a(4p-1) for all prime p except p=3. p divides a(5p-1) for prime p>3. p divides a(6p-1) for all prime except p=5. . p^2 divides a(p^2-1) for prime p>2. p^2 divides a(2p^2-1) for all prime p. p^2 divides a(3p^2-1) for all prime p. . p^3 divides a(p^3-1) for prime p>2. . p^k divides a(p^k-1) for prime p>2 and integer k>1. p^k divides a(m*p^k-1) for all prime p and integer m,k>1.
%F a(n) = Numerator[Sum[k^2/(n-k+1),{k,1,n}]]. a(n) = Numerator[HarmonicNumber[n]*(n+1)^2 - 3*n(n+1)/2]. a(n) = Numerator[A001008[n]/A002805[n]*(n+1)^2 - 3*A000217[n]].
%t Numerator[Table[Sum[k^2/(n-k+1),{k,1,n}],{n,1,50}]]
%Y Cf. A027612, A001008, A002805, A000217.
%K frac,nonn
%O 1,2
%A _Alexander Adamchuk_, Jul 26 2006
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