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%I #28 Oct 07 2023 05:09:13
%S 1,5,30,229,2165,24576,326515,4976315,85630914,1642623355,34762642871,
%T 804650577600,20224019536825,548535471681029,15969883030969470,
%U 496754110707779461,16441934503725675485,576991048929859964160,21399021201104749243099,836326710446071005267035
%N Column 2 of array in A144502.
%H Vincenzo Librandi, <a href="/A144498/b144498.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = A001515(n+1) - A001515(n).
%F a(n) = A144301(n+2) - A144301(n+1).
%F E.g.f.: (1 - 2*x + 2*x*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(1-2*x)^2. - _Sergei N. Gladkovskii_, Oct 06 2012
%F G.f.: (1-x)/(x*Q(0)) - 1/x, where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 18 2013
%F a(n) ~ 2^(n+3/2) * n^(n+1) / exp(n-1). - _Vaclav Kotesovec_, Oct 08 2013
%F G.f.: T(0)/x- 1/x, where T(k) = 1 - (k+1)*x/((k+1)*x - (1-x)^2/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 15 2013
%F (2*n-1)*a(n) = (4*n^2 + 1)*a(n-1) + (2*n+1)*a(n-2). - _G. C. Greubel_, Oct 07 2023
%p f1:=proc(n) local k; add((n+k+1)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f1(n),n=0..60)];
%t Table[Sum[(n+k+1)!/((n-k)!*k!*2^k), {k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 08 2013 *)
%o (Magma)
%o A144498:= func< n | (&+[Binomial(n,k)*Factorial(n+k+1)/(2^k*Factorial(n)): k in [0..n]]) >;
%o [A144498(n): n in [0..30]]; // _G. C. Greubel_, Oct 07 2023
%o (SageMath)
%o def A144498(n): return sum(binomial(n,k)*rising_factorial(n+1,k+1)//2^k for k in range(n+1))
%o [A144498(n) for n in range(31)] # _G. C. Greubel_, Oct 07 2023
%Y First differences of A001515 and A144301.
%K nonn
%O 0,2
%A _David Applegate_ and _N. J. A. Sloane_, Dec 13 2008
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