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A157165 Numerators of partial sums of a series related to Lebesgue's constant L(0) = 1. 5
1, 19, 734, 16294, 557407, 81759221, 1083213812, 3737624804, 1221606572113, 4687376819963, 543445163726882, 314646975551939566, 4747879086124761619, 415213127253949396153, 374983405094739446762072, 11671625151617599366571432, 11713911632041456348356827 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For the denominators see A157166.

Lebesgue's constants L(n):= (2/Pi)*int(|sin((2*n+1)*x)|/sin(x),x=0..Pi/2). (Called rho_n in the Szego reference). L(0) = 1.

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

The rationals (partial sums) R(0;n) = 3*sum(Theta(1,k)/(4*k^2-1),k=1..n) 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(Theta(1,k)/(4*k^2-1),k=1..n)), 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

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Last modified December 9 05:27 EST 2021. Contains 349627 sequences. (Running on oeis4.)