login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

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 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002084 Sinh x / cos x = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.
(Formerly M3667 N1493)
15
1, 4, 36, 624, 18256, 814144, 51475776, 4381112064, 482962852096, 66942218896384, 11394877025289216, 2336793875186479104, 568240131312188379136, 161669933656307658932224, 53204153193639888357113856, 20053432927718528320240287744 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Gandhi proves that a(n) = 1 (mod 2n+1) if 2n+1 is prime, that a(2n+1) = 4 (mod 10), and that a(2n+2) = 6 (mod 10). - Charles R Greathouse IV, Oct 16 2012

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

T. D. Noe, Table of n, a(n) for n = 0..50

J. M. Gandhi, The coefficients of sinh x/ cos x. Canad. Math. Bull. 13 1970 305-310.

Peter Luschny, An old operation on sequences: the Seidel transform

FORMULA

E.g.f.: sinh(x)/cos(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.

a(n) = Sum_{k=0..n} binomial(2n+1, 2k+1)*A000364(n-k) = Sum_{k=0..n} A103327(n, k)*A000324(n-k) = Sum_{k=0..n} (-1)^(n-k)*A104033(n, k). - Philippe Deléham, Aug 27 2005

a(n) ~ sinh(Pi/2) * 2^(2*n + 3) * (2*n + 1)! / Pi^(2*n+2). - Vaclav Kotesovec, Jul 05 2020

EXAMPLE

x + 2/3*x^3 + 3/10*x^5 + 13/105*x^7 + 163/3240*x^9 + ...

MATHEMATICA

With[{nn=30}, Take[CoefficientList[Series[Sinh[x]/Cos[x], {x, 0, nn}], x] Range[0, nn-1]!, {2, -1, 2}]] (* Harvey P. Dale, Jul 17 2012 *)

PROG

(Sage) # Generalized algorithm of L. Seidel (1877)

def A002084_list(n) :

R = []; A = {-1:0, 0:0}

k = 0; e = 1

for i in range(2*n) :

Am = 1 if e == -1 else 0

A[k + e] = 0

e = -e

for j in (0..i) :

Am += A[k]

A[k] = Am

k += e

if e == 1 : R.append(A[i//2])

return R

A002084_list(10) # Peter Luschny, Jun 02 2012

(PARI) a(n)=n++; my(v=Vec(1/cos(x+O(x^(2*n+1))))); v=vector(n, i, v[2*i-1]*(2*i-2)!); sum(g=1, n, binomial(2*n-1, 2*g-2)*v[g]) \\ Charles R Greathouse IV, Oct 16 2012

(PARI) list(n)=n++; my(v=Vec(1/cos(x+O(x^(2*n+1))))); v=vector(n, i, v[2*i-1]*(2*i-2)!); vector(n, k, sum(g=1, k, binomial(2*k-1, 2*g-2)*v[g])) \\ Charles R Greathouse IV, Oct 16 2012

CROSSREFS

Cf. A002085.

Sequence in context: A263445 A241029 A002761 * A135867 A268470 A214347

Adjacent sequences: A002081 A002082 A002083 * A002085 A002086 A002087

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

a(13)-a(15) from Andrew Howroyd, Feb 05 2018

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 10:26 EST 2022. Contains 358656 sequences. (Running on oeis4.)