A154603 Binomial transform of reduced tangent numbers (A002105).

1, 1, 2, 4, 11, 31, 110, 400, 1757, 7861, 41402, 220540, 1358183, 8405203, 59340710, 418689544, 3335855897, 26440317193, 234747589106, 2065458479476, 20224631361251, 195625329965671, 2094552876276830, 22092621409440256

Offset: 0

Author

Paul Barry, Jan 12 2009

Keywords

• easy
• nonn

Comments

Hankel transform is A154604.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A002105, A154604.

Programs

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

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 *)

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

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

Extensions

Typo in e.g.f. fixed by Vaclav Kotesovec, Oct 02 2013