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%I M5229 N2276 #35 Dec 22 2023 08:35:19
%S 31,3661,1217776,929081776,1413470290176,3878864920694016,
%T 17810567950611972096,129089983180418186674176,
%U 1409795030885143760732160000,22335321387514981111936450560000,497400843208278958640564703068160000,15161356456130244705175927906904309760000
%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 Mircea Merca, <a href="https://dx.doi.org/10.1007/s10998-014-0034-3">Some experiments with complete and elementary symmetric functions</a>, Periodica Mathematica Hungarica, 69 (2014), 182-189.
%F a(n) = (n + 1)!^4/480*(20*Psi(n + 2)^4 + 80*gamma*Psi(n + 2)^3 - 120*Psi(n + 2)^2*Psi(1, n + 2) + 20*Pi^2*Psi(n + 2)^2 + 120*gamma^2*Psi(n + 2)^2 - 240*gamma*Psi(n + 2)*Psi(1, n + 2) + 80*Psi(n + 2)*Psi(2, n + 2) + 60*Psi(1, n + 2)^2 + 40*gamma*Pi^2*Psi(n + 2) + 160*Zeta(3)*Psi(n + 2) + 80*gamma^3*Psi(n + 2) - 20*Pi^2*Psi(1, n + 2) - 120*gamma^2*Psi(1, n + 2) + 80*gamma*Psi(2, n + 2) - 20*Psi(3, n + 2) + 160*gamma*Zeta(3) + 3*Pi^4 + 20*gamma^4 + 20*gamma^2*Pi^2). - _Vladeta Jovovic_, Aug 10 2002
%F a(n) = (n+1)!^4 * Sum[i=1..n+1, Sum[j=1..i, Sum[k=1..j, Sum[l=1..k, 1/(ijkl) ]]].
%F a(n) = ((n+1)!)^4 * sum((-1)^(k+1)*C(n+1,k)/k^4,k=1..n+1). - _Sean A. Irvine_, Mar 29 2012
%t a[n_] := -(Factorial[n + 1]^4)*Sum[(-1)^k Binomial[n + 1, k]/k^4, {k, 1, n + 1}];Table[a[n],{n,14}] (* _James C. McMahon_, Dec 12 2023 *)
%o (PARI) a(n)=-(n+1)!^4*sum(k=1,n+1,(-1)^k*binomial(n+1,k)/k^4) \\ _Charles R Greathouse IV_, Mar 29 2012
%Y Column 4 in triangle A008969.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Aug 10 2002
%E a(11)-a(12) from _James C. McMahon_, Dec 12 2023
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