A154603 - OEIS

%I #26 Sep 20 2024 05:41:43

%S 1,1,2,4,11,31,110,400,1757,7861,41402,220540,1358183,8405203,

%T 59340710,418689544,3335855897,26440317193,234747589106,2065458479476,

%U 20224631361251,195625329965671,2094552876276830,22092621409440256

%N Binomial transform of reduced tangent numbers (A002105).

%C Hankel transform is A154604.

%H Vincenzo Librandi, <a href="/A154603/b154603.txt">Table of n, a(n) for n = 0..300</a>

%F 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);

%F E.g.f.: exp(x)*(sec(x/sqrt(2))^2);

%F 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

%F 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

%F 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

%F 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

%F 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

%t 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 *)

%o (Magma)

%o A002105:= func< n | (-1)^(n+1)*2^n*(4^n - 1)*Bernoulli(2*n)/n >;

%o b:= func< n | (n mod 2) eq 0 select A002105(Floor(n/2)+1) else 0 >;

%o A154603:= func< n | (&+[Binomial(n,k)*b(k): k in [0..n]]) >;

%o [A154603(n): n in [0..30]]; // _G. C. Greubel_, Sep 20 2024

%o (SageMath)

%o def A002105(n): return (-1)^(n+1)*2^n*(4^n -1)*bernoulli(2*n)/n

%o def b(n): return A002105(n//2 +1) if n%2==0 else 0

%o def A154603(n): return sum(binomial(n,k)*b(k) for k in range(n+1))

%o [A154603(n) for n in range(31)] # _G. C. Greubel_, Sep 20 2024

%Y Cf. A002105, A154604.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Jan 12 2009

%E Typo in e.g.f. fixed by _Vaclav Kotesovec_, Oct 02 2013