cp's OEIS Frontend

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

%I A060944 #38 Feb 16 2025 08:32:44
%S A060944 1,9,130,2900,93576,4141872,241353792,17929776384,1655071418880,
%T A060944 185914776960000,24978180045312000,3955930130221056000,
%U A060944 729464836964806656000,154952762244805582848000,37566943754471090749440000,10310706109241121091092480000
%N A060944 a(n) = n!^2 * Sum_{k=1..n} Sum_{j=1..k} 1/j^2.
%C A060944 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 A060944 Harry J. Smith, <a href="/A060944/b060944.txt">Table of n, a(n) for n = 1..100</a>
%H A060944 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%F A060944 From _Alexander Adamchuk_, Oct 27 2004: (Start)
%F A060944 a(n) = (n!)^2 * Sum_{k=0..n-1} (k+1)/(n-k)^2.
%F A060944 a(n) = (n!)^2 * Sum_{k=1..n} HarmonicNumber(k, 2), where HarmonicNumber(k, 2) = A007406(k) / A007407(k). (End)
%F A060944 Sum_{n>=1} a(n) * x^n / (n!)^2 = polylog(2,x) / (1 - x)^2. - _Ilya Gutkovskiy_, Jul 15 2020
%e A060944 a(3) = 6^2 *(1 + (1 + 1/2^2) + (1 + 1/2^2 + 1/3^2)) = 130.
%p A060944 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 A060944 Table[(n!)^2*Sum[(k+1)/(n-k)^2, {k, 0, n-1}], {n, 1, 10}]
%o A060944 (PARI) a(n)={n!^2 * sum(k=1, n, sum(j=1, k, 1/j^2))} \\ _Harry J. Smith_, Jul 15 2009
%o A060944 (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 A060944 (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 A060944 Cf. A001705, A007406, A007407.
%K A060944 easy,nonn
%O A060944 1,2
%A A060944 _Leroy Quet_, May 07 2001