A001237 Differences of reciprocals of unity. (Formerly M5229 N2276)

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

Offset

1,1

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

Table of n, a(n) for n=1..12.

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

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

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 *)

Prog

(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

Crossrefs

Column 4 in triangle A008969.

Sequence in context: A218661 A183783 A072913 * A289397 A177465 A187755

Adjacent sequences: A001234 A001235 A001236 * A001238 A001239 A001240

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Aug 10 2002

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

Status

approved