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A060944 a(n) = n!^2 * Sum_{k=1..n} Sum_{j=1..k} 1/j^2. 1

%I #37 Apr 09 2021 09:17:13

%S 1,9,130,2900,93576,4141872,241353792,17929776384,1655071418880,

%T 185914776960000,24978180045312000,3955930130221056000,

%U 729464836964806656000,154952762244805582848000,37566943754471090749440000,10310706109241121091092480000

%N a(n) = n!^2 * Sum_{k=1..n} Sum_{j=1..k} 1/j^2.

%C Sum of generalized harmonic numbers squared multiplied by (n!)^2. agenh(n) = Sum_{k=1..n} HarmonicNumber(k, 2), where HarmonicNumber(n, j) = Sum_{k = 1..n} 1/k^j. - _Alexander Adamchuk_, Oct 27 2004

%H Harry J. Smith, <a href="/A060944/b060944.txt">Table of n, a(n) for n = 1..100</a>

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

%F From _Alexander Adamchuk_, Oct 27 2004: (Start)

%F a(n) = (n!)^2 * Sum_{k=0..n-1} (k+1)/(n-k)^2.

%F a(n) = (n!)^2 * Sum_{k=1..n} HarmonicNumber(k, 2), where HarmonicNumber(k, 2) = A007406(k) / A007407(k). (End)

%F Sum_{n>=1} a(n) * x^n / (n!)^2 = polylog(2,x) / (1 - x)^2. - _Ilya Gutkovskiy_, Jul 15 2020

%e a(3) = 6^2 *(1 + (1 + 1/2^2) + (1 + 1/2^2 + 1/3^2)) = 130.

%p A060944:= n-> (n!)^2*add((1+j)/(n-j)^2, j=0..n-1); seq(A060944(n), n=1..15); # _G. C. Greubel_, Apr 09 2021

%t Table[(n!)^2*Sum[(k+1)/(n-k)^2, {k, 0, n-1}], {n, 1, 10}]

%o (PARI) a(n)={n!^2 * sum(k=1, n, sum(j=1, k, 1/j^2))} \\ _Harry J. Smith_, Jul 15 2009

%o (Magma) [(Factorial(n))^2*(&+[(1+j)/(n-j)^2: j in [0..n-1]]): n in [1..15]]; // _G. C. Greubel_, Apr 09 2021

%o (Sage) [(factorial(n))^2*sum((1+j)/(n-j)^2 for j in (0..n-1)) for n in (1..15)] # _G. C. Greubel_, Apr 09 2021

%Y Cf. A001705, A007406, A007407.

%K easy,nonn

%O 1,2

%A _Leroy Quet_, May 07 2001

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)