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A154603 Binomial transform of reduced tangent numbers (A002105).

%I A154603 #26 Sep 20 2024 05:41:43
%S A154603 1,1,2,4,11,31,110,400,1757,7861,41402,220540,1358183,8405203,
%T A154603 59340710,418689544,3335855897,26440317193,234747589106,2065458479476,
%U A154603 20224631361251,195625329965671,2094552876276830,22092621409440256
%N A154603 Binomial transform of reduced tangent numbers (A002105).
%C A154603 Hankel transform is A154604.
%H A154603 Vincenzo Librandi, <a href="/A154603/b154603.txt">Table of n, a(n) for n = 0..300</a>
%F A154603 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 A154603 E.g.f.: exp(x)*(sec(x/sqrt(2))^2);
%F A154603 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 A154603 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 A154603 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 A154603 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 A154603 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 A154603 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 A154603 (Magma)
%o A154603 A002105:= func< n | (-1)^(n+1)*2^n*(4^n - 1)*Bernoulli(2*n)/n >;
%o A154603 b:= func< n | (n mod 2) eq 0 select A002105(Floor(n/2)+1) else 0 >;
%o A154603 A154603:= func< n | (&+[Binomial(n,k)*b(k): k in [0..n]]) >;
%o A154603 [A154603(n): n in [0..30]]; // _G. C. Greubel_, Sep 20 2024
%o A154603 (SageMath)
%o A154603 def A002105(n): return (-1)^(n+1)*2^n*(4^n -1)*bernoulli(2*n)/n
%o A154603 def b(n): return A002105(n//2 +1) if n%2==0 else 0
%o A154603 def A154603(n): return sum(binomial(n,k)*b(k) for k in range(n+1))
%o A154603 [A154603(n) for n in range(31)] # _G. C. Greubel_, Sep 20 2024
%Y A154603 Cf. A002105, A154604.
%K A154603 easy,nonn
%O A154603 0,3
%A A154603 _Paul Barry_, Jan 12 2009
%E A154603 Typo in e.g.f. fixed by _Vaclav Kotesovec_, Oct 02 2013