A002547 Numerator of the n-th harmonic number H(n) divided by (n+1); a(n) = A001008(n) / ((n+1)*A002805(n)).

1, 1, 11, 5, 137, 7, 363, 761, 7129, 671, 83711, 6617, 1145993, 1171733, 1195757, 143327, 42142223, 751279, 275295799, 55835135, 18858053, 830139, 444316699, 269564591, 34052522467, 34395742267, 312536252003, 10876020307, 9227046511387, 300151059037

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

Numerators of coefficients for numerical differentiation.

Examples

			H(n) = Sum_{k=1..n} 1/k, begins 1, 3/2, 11/6, 25/12, ... so H(n)/(n+1) begins 1/2, 1/2, 11/24, 5/12, ....
a(4) = numerator(H(4)/(4+1)) = 5.
		

References

• W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
• A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924.
• 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

• M. F. Hasler, Table of n, a(n) for n = 1..2000 (first 700 terms from Alois P. Heinz)
• W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
• A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924. [Annotated scanned copy]
• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Cf. A002548, A001008, A002805.

Programs

GAP

List([1..35], n-> NumeratorRat(Sum([1..n], k-> 1/k)/(n+1))); # G. C. Greubel, Jul 03 2019

Magma

[Numerator(HarmonicNumber(n)/(n+1)): n in [1..35]]; // G. C. Greubel, Jul 03 2019

Maple

H := proc(a, b) option remember; local m, p, q, r, s;
if b - a <= 1 then return 1, a fi; m := iquo(a + b, 2);
p, q := H(a, m); r, s := H(m, b); p*s + q*r, q*s; end:
A002547 := proc(n) H(1, n+1); numer(%[1]/(%[2]*(n+1))) end:
seq(A002547(n), n=1..30); # Peter Luschny, Jul 11 2019

Mathematica

a[n_]:= Numerator[HarmonicNumber[n]/(n+1)]; Table[a[n], {n, 35}] (* modified by G. C. Greubel, Jul 03 2019 *)

PARI

h(n) = sum(k=1, n, 1/k);
vector(35, n, numerator(h(n)/(n+1))) \\ G. C. Greubel, Jul 03 2019

PARI

A002547(n)=numerator(A001008(n)/(n+1)) \\ M. F. Hasler, Jul 03 2019

Sage

[numerator(harmonic_number(n)/(n+1)) for n in (1..35)] # G. C. Greubel, Jul 03 2019

Formula

G.f.: (-log(1-x))^2 (for fractions A002547(n)/A002548(n)). - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

A002547(n)/A002548(n) = 2*Stirling_1(n+2, 2)(-1)^n/(n+2)! - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

Numerator of u(n) = Sum_{k=1..n-1} 1/(k*(n-k)) (u(n) is asymptotic to 2*log(n)/n). - Benoit Cloitre, Apr 12 2003; corrected by Istvan Mezo, Oct 29 2012

a(n) = numerator of 2*Integral_{0..1} x^(n+1)*log(x/(1-x)) dx. - Groux Roland, May 18 2011

a(n) = numerator of A001008(n)/(n+1), since A001008(n)/A002805(n) are already in lowest terms. - M. F. Hasler, Jul 03 2019

Extensions

More terms from Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

Simpler definition from Alexander Adamchuk, Oct 31 2004

Offset corrected by Gary Detlefs, Sep 08 2011

Definition corrected by M. F. Hasler, Jul 03 2019