cp's OEIS Frontend

A165433 A transform of the double factorial numbers A001147.

%I A165433 #17 Apr 18 2024 10:01:10
%S A165433 1,1,2,3,7,14,39,97,308,897,3139,10304,38997,140893,570002,2230599,
%T A165433 9567979,40091222,181203603,805962157,3819522284,17912075229,
%U A165433 88646095447,435959031488,2245454002137,11530035000169,61627679281154
%N A165433 A transform of the double factorial numbers A001147.
%C A165433 Hankel transform is A000178.
%H A165433 G. C. Greubel, <a href="/A165433/b165433.txt">Table of n, a(n) for n = 0..800</a>
%F A165433 G.f.: 1/(1-x-x^2-2x^4/(1-x-5x^2-12x^4/(1-x-9x^2-30x^4/(1-x-13x^2-56x^4/(1-.... (continued fraction);
%F A165433 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(2k)!/(k!*2^k).
%F A165433 Conjecture: 2*a(n) -3*a(n-1) +(3-2*n)*a(n-2) +(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Nov 14 2011
%F A165433 G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 29 2013
%F A165433 a(n) ~ 2^(-1/2) * exp(sqrt(n)/2 - n/2 + 1/16) * n^(n/2) * (1 + 121/(192*sqrt(n))). - _Vaclav Kotesovec_, Apr 18 2024
%p A165433 a:=proc(n) add(binomial(n-k,k)*factorial(2*k)/(factorial(k)*2^k),k=0..floor(n/2)) end proc: seq(a(n),n=0..30); # _Muniru A Asiru_, Oct 20 2018
%t A165433 Table[Sum[Binomial[n-k, k]*(2*k)!/(k!*2^k), {k, 0, Floor[n/2]}], {n,0, 30}] (* _G. C. Greubel_, Oct 20 2018 *)
%o A165433 (PARI) vector(30, n, n--; sum(k=0, floor(n/2), binomial(n-k,k)*(2*k)!/(k!*2^k))) \\ _G. C. Greubel_, Oct 20 2018
%o A165433 (Magma) [(&+[Binomial(n-k,k)*Factorial(2*k)/(Factorial(k)*2^k): k in [0.. Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Oct 20 2018
%K A165433 easy,nonn
%O A165433 0,3
%A A165433 _Paul Barry_, Sep 18 2009