A157165 - OEIS

%I #24 Feb 16 2025 08:33:09

%S 1,19,734,16294,557407,81759221,1083213812,3737624804,1221606572113,

%T 4687376819963,543445163726882,314646975551939566,4747879086124761619,

%U 415213127253949396153,374983405094739446762072,11671625151617599366571432,11713911632041456348356827

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

%C For the denominators see A157166.

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

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

%C 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).

%C 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).

%H Wolfdieter Lang, <a href="/A157165/a157165.txt">Sequences related to Lebesgue's constants.</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LebesgueConstants.html">Lebesgue Constants</a>

%H G. Szego, <a href="http://dx.doi.org/10.1007/BF01667326">Über die Lebesgueschen Konstanten bei den Fourierschen Reihen</a>, Math. Z. 9 (1921) 163-166.

%F 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.

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

%t 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 *)

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

%K nonn,frac,easy,changed

%O 1,2

%A _Wolfdieter Lang_, Oct 16 2009, Nov 24 2009