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A000318 Generalized tangent numbers d(4,n).
(Formerly M3713 N1517)
6
4, 128, 16384, 4456448, 2080374784, 1483911200768, 1501108249821184, 2044143848640217088, 3605459138582973251584, 7995891855149741436305408, 21776918737280678860353961984, 71454103701490016776039304265728, 278008871543597996197497752082448384 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Padé approximants.
FORMULA
a(n) = 2^(4n-2) * A000182(n).
The g.f. has the following continued fraction expansion: g.f. = [4, b(0), c(0), b(1), c(1), b(2), c(2), ...] where b(n) = (Sum_{k=0..n} 1/(2*k+1))^2 / (128*(n+1)*x), c(n) = -4/((2*n+3)*(Sum_{k=0..n} 1/(2*k+1))*(Sum_{k=0..n+1} 1/(2*k+1))) and each convergent of this continued fraction is a Padé approximant of the Maclaurin series Sum_{k>=1} a(n)*x^(n-1). - Thomas Baruchel, Oct 19 2005
a(n) = (2*n-1)!*[x^(2*n-1)](sec(4*x)*sin(4*x)). - Peter Luschny, Nov 21 2021
MAPLE
egf := sec(4*x)*sin(4*x): ser := series(egf, x, 26):
seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..12); # Peter Luschny, Nov 21 2021
MATHEMATICA
nn = 30; t = Rest@Union[Range[0, nn - 1]! CoefficientList[Series[Tan[x], {x, 0, nn}], x]]; t2 = t*2^Range[2, 2*nn, 4] (* T. D. Noe, Jun 19 2012 *)
CROSSREFS
Sequence in context: A013823 A321233 A130318 * A229385 A141367 A141368
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 03 2000
STATUS
approved

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)