This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A024235 E.g.f. tan(x)*sin(x)/2 (even powers only). 8
 0, 1, 2, 31, 692, 25261, 1351382, 99680491, 9695756072, 1202439837721, 185185594118762, 34674437196568951, 7757267081778543452, 2043536254646561946181, 626129820701814932734142, 220771946624511552276841411, 88759695789769644718332394832 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Peter Bala, Nov 10 2016: (Start) This sequence gives the coefficients in an asymptotic expansion related to the constant Pi/8. Recall the Madhava-Gregory-Leibniz series Pi/4 = Sum_{k = 1..inf} (-1)^(k-1)/(2*k - 1). Borwein et al. gave an asymptotic expansion for the tails of this series: Pi/2 - 2*Sum_{k = 1..N/2} (-1)^(k-1)/(2*k - 1) ~ 1/N - 1/N^3 + 5/N^5 - 61/N^7 + ..., where N is an integer divisible by 4 and the sequence of unsigned coefficients [1, 1, 5, 61,...] is the sequence of Euler numbers A000364. Similarly, we have the series representation Pi/8 = Sum_{k = 1..inf} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)): using the approach of Borwein et al. we can show the associated asymptotic expansion for the tails of the series is Pi/4 - 2*Sum_{k = 1..N/2} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)) ~ -1/N^3 + 2/N^5 - 31/N^7 + 692/N^9 - ..., where N is divisible by 4 and where the sequence of unsigned coefficients [1, 2, 31, 692,...] forms the present sequence. A numerical example is given below. Cf. A278080 and A278195. (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..50 J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687. Eric Weisstein's World of Mathematics, MathWorld: Euler Polynomial FORMULA G.f.: 1/2*(G(0) - 1/(1+x)) where G(k) = 1 - x*(2*k+1)^2/(1 - x*(2*k+2)^2/G(k+1) );  (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 09 2013 a(n) ~ (2*n)! * (2/Pi)^(2*n+1). - Vaclav Kotesovec, Jan 23 2015 From Peter Bala, Nov 10 2016: (Start) a(n) = 1/2*(A000364(n) - (-1)^n). a(n) = 1/8*(-4)^n*( -E(2*n,3/2) + 2*E(2*n,1/2) - E(2*n,-1/2) ), where E(n,x) is the Euler polynomial of order n. G.f. 1/2!*sin^2(x)/cos(x) = x^2/2! + 2*x^4/4! + 31*x^6/6! + 692*x^8/8! + .... O.g.f. for a signed version of the sequence: Sum_{n >= 0} ( 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n, k)/((1 - (2*k - 1)*x)*(1 - (2*k + 1)*x)*(1 - (2*k + 3)*x)) ) = 1 - 2*x^2 + 31*x^4 - 692*x^6 + .... (End) EXAMPLE tan(x)*sin(x)/2 = 1/2*x^2 + 1/12*x^4 + 31/720*x^6 + 173/10080*x^8 + ... From Peter Bala, Nov 10 2016: (Start) Asymptotic expansion at N = 100000. The truncated series 2*Sum_{k = 1..N/2} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)) = 0.78539816339744(9)309615660(6)4581987(603) 104929(1657)84377... to 50 digits. The bracketed digits show where this decimal expansion differs from that of Pi/4. The numbers -1, 2, -31, 692 must be added to the bracketed numbers to give the correct decimal expansion to 50 digits: Pi/4 = 0.78539816339744(8)309615660(8)4581987(572)104929(2349)84377.... (End) MAPLE A000364 := proc(n)    abs(euler(2*n)); end proc: seq(1/2*(A000364(n) - (-1)^n), n = 0..20); # Peter Bala, Nov 10 2016 MATHEMATICA With[{nn=30}, Take[CoefficientList[Series[Tan[x]*Sin[x]/2, {x, 0, nn}], x]Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Apr 27 2012 *) CROSSREFS Cf. A009744, A000364, A004174, A019675, A278080, A278195. Sequence in context: A057692 A058244 A245051 * A010789 A224863 A263075 Adjacent sequences:  A024232 A024233 A024234 * A024236 A024237 A024238 KEYWORD nonn,easy AUTHOR EXTENSIONS Extended and signs tested Mar 15 1997. More terms from Harvey P. Dale, Apr 27 2012 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 October 22 19:53 EDT 2019. Contains 328319 sequences. (Running on oeis4.)