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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
a(13)-a(15) from Andrew Howroyd, Feb 05 2018
STATUS
approved