A002084 Sinh(x) / cos(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.

1, 4, 36, 624, 18256, 814144, 51475776, 4381112064, 482962852096, 66942218896384, 11394877025289216, 2336793875186479104, 568240131312188379136, 161669933656307658932224, 53204153193639888357113856, 20053432927718528320240287744

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

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

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.

Crossrefs

Cf. A002085.

Programs

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 *)

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

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

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

Extensions

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