The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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