A003701 Expansion of e.g.f. exp(x)/cos(x).

1, 1, 2, 4, 12, 36, 152, 624, 3472, 18256, 126752, 814144, 6781632, 51475776, 500231552, 4381112064, 48656756992, 482962852096, 6034272215552, 66942218896384, 929327412759552, 11394877025289216, 174008703107274752, 2336793875186479104, 38928735228629389312

Offset: 0

Author

R. H. Hardin

Keywords

• nonn

Comments

Binomial transform of A000364 (with interpolated zeros). Hankel transform is A055209. - Paul Barry, Jan 12 2009

Examples

			G.f. = 1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 152*x^6 + 624*x^7 + 3472*x^8 + ...
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..485 (first 101 terms from T. D. Noe)
• T. Chow, Fair permutations and random k-sets, Problem 11523, Amer. Math. Monthly 117 (October 2010), 741; solution by Jim Simons, Amer. Math. Monthly 119 (November 2012), 801-803.
• J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Crossrefs

Cf. A009739, A062272, A062161.

Bisections are A000795 and A002084.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)/Cos(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 14 2018

Maple

G(x):= exp(x)*sec(x): f[0]:=G(x): for n from 1 to 54 do f[n]:= diff(f[n-1],x) od: x:=0: seq(f[n], n=0..22); # Zerinvary Lajos, Apr 05 2009
# second Maple program:
b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> add(`if`(j::odd, 0, b(j, 0)*binomial(n, j)), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 12 2024

Mathematica

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x ] / Cos[x], {x, 0, n}]] (* Michael Somos, Jun 06 2012 *)

PARI

x='x+O('x^66); Vec(serlaplace(exp(x)/cos(x))) \\ Joerg Arndt, May 07 2013

Formula

G.f.: 1/(1-x-x^2/(1-x-4x^2/(1-x-9x^2/(1-x-16x^2.... (continued fraction). - Paul Barry, Jan 12 2009

E.g.f.: exp(x)*sec(x). - Zerinvary Lajos, Apr 05 2009

E.g.f.: 1+x/H(0); H(k)=4k+1-x+x^2*(4k+1)/((2k+1)*(4k+3)-x^2+x*(2k+1)*(4k+3)/(2k+2-x+x*(2k+2)/H(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011

G.f.: 1/G(0) where G(k)= 1 - 2*x*(k+1)/(1 + 1/(1 + 2*x*(k+1)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 20 2012

G.f.: -1/x/Q(0), where Q(k)= 1 - 1/x - (k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 26 2013

G.f.: (1-x)/Q(0), where Q(k)= (1-x)^2 - (1-x)^2*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013

a(n) ~ n! * ((-1)^n*exp(-Pi/2) + exp(Pi/2)) *(2/Pi)^(n+1). - Vaclav Kotesovec, Oct 08 2013

G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)/( x*(2*k+1) - 1/(1 + x*(2*k+1)/( x*(2*k+1) + 1/(1 - x*(2*k+2)/( x*(2*k+2) - 1/(1 + x*(2*k+2)/( x*(2*k+2) + 1/Q(k+1) ))))))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013

G.f.: Q(0)/(1-x), where Q(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 - (1-x)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013

Extensions

Extended and reformatted 03/97.