OFFSET
0,3
COMMENTS
Hankel transform is A154604.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
FORMULA
G.f: 1/(1-x-x^2/(1-x-3x^2/(1-x-6x^2/(1-x-10x^2/(1-x-15x^2..... (continued fraction);
E.g.f.: exp(x)*(sec(x/sqrt(2))^2);
G.f.: 1/(x*Q(0)), where Q(k)= 1/x - 1 - (k+1)*(k+2)/2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 26 2013
G.f.: 1/Q(0), where Q(k)= 1 - x - 1/2*x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
a(n) ~ n! * 2^(2+n/2)*n*(exp(sqrt(2)*Pi)+(-1)^n) / (Pi^(n+2)*exp(Pi/sqrt(2))). - Vaclav Kotesovec, Oct 02 2013
G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 14 2013
a(n) = Sum_{k=0..n} binomial(n,k)*b(k), where b(n) = A002105((n+2)/2) if n mod 2 = 0 otherwise b(n) = 0. - G. C. Greubel, Sep 20 2024
MATHEMATICA
With[{nn=30}, CoefficientList[Series[Exp[x]Sec[x/Sqrt[2]]^2, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, May 30 2013 *)
PROG
(Magma)
A002105:= func< n | (-1)^(n+1)*2^n*(4^n - 1)*Bernoulli(2*n)/n >;
b:= func< n | (n mod 2) eq 0 select A002105(Floor(n/2)+1) else 0 >;
A154603:= func< n | (&+[Binomial(n, k)*b(k): k in [0..n]]) >;
[A154603(n): n in [0..30]]; // G. C. Greubel, Sep 20 2024
(SageMath)
def A002105(n): return (-1)^(n+1)*2^n*(4^n -1)*bernoulli(2*n)/n
def b(n): return A002105(n//2 +1) if n%2==0 else 0
def A154603(n): return sum(binomial(n, k)*b(k) for k in range(n+1))
[A154603(n) for n in range(31)] # G. C. Greubel, Sep 20 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jan 12 2009
EXTENSIONS
Typo in e.g.f. fixed by Vaclav Kotesovec, Oct 02 2013
STATUS
approved