A157165 Numerators of partial sums of a series related to Lebesgue's constant L(0) = 1.

1, 19, 734, 16294, 557407, 81759221, 1083213812, 3737624804, 1221606572113, 4687376819963, 543445163726882, 314646975551939566, 4747879086124761619, 415213127253949396153, 374983405094739446762072, 11671625151617599366571432, 11713911632041456348356827

Offset

1,2

Comments

For the denominators see A157166.

Lebesgue's constants L(n) := (2/Pi)*Integral_{x=0..Pi/2}

|sin((2*n+1)*x)|/sin(x) dx. (Called rho_n in the Szego reference.) L(0) = 1.

1 = L(0) = (16/Pi^2)*Sum_{k>=1} Theta(1,k)/(4*k^2-1) with Theta(1,k) = Sum_{j=1..k} 1/(2*j-1) = Integral_{x=0..Pi/2} sin(k*x)^2/sin(x) dx (see Szego reference formula (R), p. 165 and the line before this).

The rationals (partial sums) R(0;n) = 3*Sum_{k=1..n} Theta(1,k)/(4*k^2-1) give (in lowest terms) A157165(n)/A157166(n). The sequence {R(0;n)/3} converges slowly to (Pi^2)/16, approximately 0.6168502752 because L(0)=1 (see the W. Lang link for R(0;10^n)/3 for n=0..4).

Links

Table of n, a(n) for n=1..17.

Wolfdieter Lang, Sequences related to Lebesgue's constants.

Eric Weisstein's World of Mathematics, Lebesgue Constants

G. Szego, Über die Lebesgueschen Konstanten bei den Fourierschen Reihen, Math. Z. 9 (1921) 163-166.

Formula

a(n) = numerator(R(0;n)) = numerator(3*Sum_{k=1..n} Theta(1,k)/(4*k^2-1)), n >= 1, with Theta(1,k) defined above.

Example

Rationals R(0;n) = A157165(n)/ A157166(n): [1, 19/15, 734/525, 16294/11025, 557407/363825, 81759221/52026975, ...].

Mathematica

theta[1, k_] := Sum[1 / (2j-1), {j, 1, k}]; a[n_] := Numerator[ 3*Sum[theta[1, k]/(4k^2 - 1), {k, 1, n}]]; Table[a[n], {n, 1, 17}]  (* Jean-François Alcover, Nov 03 2011, after given formula *)

Crossrefs

A157167/A157168 for 45*((Pi^2)/16)*L(1) partial sums.

Sequence in context: A180990 A041687 A041684 * A201708 A280625 A183441

Adjacent sequences: A157162 A157163 A157164 * A157166 A157167 A157168

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Oct 16 2009, Nov 24 2009

Status

approved