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A002084 Sinh x / cos x = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.
(Formerly M3667 N1493)
9
1, 4, 36, 624, 18256, 814144, 51475776, 4381112064, 482962852096, 66942218896384, 11394877025289216, 2336793875186479104, 568240131312188379136 (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

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

STATUS

approved

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Last modified November 23 20:23 EST 2017. Contains 295141 sequences.