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%I M4993 N2149 #30 Apr 28 2020 11:51:37
%S 15,575,46760,6998824,1744835904,673781602752,381495483224064,
%T 303443622431870976,327643295527342080000,466962174913357393920000,
%U 858175477913267353681920000,1993920215002599923346309120000,5758788816015998806424467537920000
%N Differences of reciprocals of unity.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A001236/b001236.txt">Table of n, a(n) for n = 1..70</a>
%H Mircea Merca, <a href="https://www.researchgate.net/publication/264664262_Some_experiments_with_complete_and_elementary_symmetric_functions">Some experiments with complete and elementary symmetric functions</a>, Periodica Mathematica Hungarica, 69 (2014), 182-189.
%F a(n) = (n+1)!^3 * Sum_{i=1..n+1} Sum_{j=1..i} Sum_{k=1..j} 1/(i*j*k).
%F From _Vladeta Jovovic_, Jan 30 2005: (Start)
%F a(n) = (n!^3/6)*(H(n, 1)^3+3*H(n, 1)*H(n, 2)+2*H(n, 3)), where H(n, m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.
%F a(n) = (n!^3/6)*((Psi(n+1)+gamma)^3+3*(Psi(n+1)+gamma)*(-Psi(1, n+1)+1/6*Pi^2)+Psi(2, n+1)+2*Zeta(3)).
%F a(n) = n!^3*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k^3.
%F Sum_{n>=0} a(n)*x^n/n!^3 = polylog(3, x/(x-1))/(x-1). (offset 2). (End)
%p a:= n-> (n+1)!^3* add((-1)^(k+1) *binomial(n+1, k)/ k^3, k=1..n+1):
%p seq (a(n), n=1..15); # _Alois P. Heinz_, Sep 05 2008
%t h = HarmonicNumber; a[n_] := ((n+1)!^3/6)*(h[n+1, 1]^3 + 3*h[n+1, 1]*h[n+1, 2] + 2*h[n+1, 3]); Table[a[n], {n, 1, 15}] (* _Jean-François Alcover_, Feb 26 2015, after _Vladeta Jovovic_ *)
%Y Column 3 in triangle A008969.
%Y Cf. A000254, A000424, A001237, A001238.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Alois P. Heinz_, Sep 05 2008
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