A001237 Differences of reciprocals of unity.

31, 3661, 1217776, 929081776, 1413470290176, 3878864920694016, 17810567950611972096, 129089983180418186674176, 1409795030885143760732160000, 22335321387514981111936450560000, 497400843208278958640564703068160000, 15161356456130244705175927906904309760000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Crossrefs

Column 4 in triangle A008969.

Programs

Mathematica

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 *)

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

Formula

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

a(n) = (n+1)!^4 * Sum_{i=1..n+1} Sum_{j=1..i} Sum_{k=1..j} Sum_{l=1..k} 1/(ijkl).

a(n) = (n+1)!^4 * Sum_{k=1..n+1} (-1)^(k+1)*C(n+1,k)/k^4. - Sean A. Irvine, Mar 29 2012

Extensions

More terms from Vladeta Jovovic, Aug 10 2002

a(11)-a(12) from James C. McMahon, Dec 12 2023