A001236 Differences of reciprocals of unity. (Formerly M4993 N2149)

15, 575, 46760, 6998824, 1744835904, 673781602752, 381495483224064, 303443622431870976, 327643295527342080000, 466962174913357393920000, 858175477913267353681920000, 1993920215002599923346309120000, 5758788816015998806424467537920000

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

Alois P. Heinz, Table of n, a(n) for n = 1..70

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

Formula

a(n) = (n+1)!^3 * Sum_{i=1..n+1} Sum_{j=1..i} Sum_{k=1..j} 1/(i*j*k).

From Vladeta Jovovic, Jan 30 2005: (Start)

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.

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)).

a(n) = n!^3*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k^3.

Sum_{n>=0} a(n)*x^n/n!^3 = polylog(3, x/(x-1))/(x-1). (offset 2). (End)

Maple

a:= n-> (n+1)!^3* add((-1)^(k+1) *binomial(n+1, k)/ k^3, k=1..n+1):
seq (a(n), n=1..15);  # Alois P. Heinz, Sep 05 2008

Mathematica

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 *)

Crossrefs

Column 3 in triangle A008969.

Cf. A000254, A000424, A001237, A001238.

Sequence in context: A027462 A329122 A027534 * A263887 A183546 A179895

Adjacent sequences: A001233 A001234 A001235 * A001237 A001238 A001239

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Alois P. Heinz, Sep 05 2008

Status

approved