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a(n) = 2^floor(n/2) n! [x^n] exp(x+x^2/4), where [x^n] f(x) denotes the coefficient of x^n in the expansion of f(x).
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%I #39 Nov 30 2019 04:53:49

%S 1,1,3,5,19,39,173,407,2025,5281,28787,81597,479851,1459015,9156093,

%T 29582303,196506001,669822849,4680247715,16737058997,122398824579,

%U 457140004519,3484655325197,13541735424615,107230543328761,432232193519521,3545227970258067

%N a(n) = 2^floor(n/2) n! [x^n] exp(x+x^2/4), where [x^n] f(x) denotes the coefficient of x^n in the expansion of f(x).

%H Vincenzo Librandi, <a href="/A242818/b242818.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{k=0..n} (s(n,k)*(-1)^n-k)*b(k)/2^(ceiling(n/2)-k) where the s(n,k) are the (signless) Stirling numbers of the first kind, and the b(n) are the Bell numbers.

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*(-1)^k*2^(floor(n/2)-2*k)*((2*k)!/k!)*i(n-2*k), where the i(n) are the involution numbers A000085.

%F Recurrence: a(n+2) = 2^even(n)*a(n+1) + (n+1)*a(n), where even(n) = 1 when n is even and even(n) = 0 otherwise.

%F G.f.: Sum_{n>=0} a(n)*x^n/(2^floor(n/2)*n!) = exp(x+x^2/4).

%F a(n) ~ 2^((-3 + (-1)^n)/4) * n^(n/2) / exp(n/2 + 1/2 - sqrt(2*n)) * (1+sqrt(2/n)/3). - _Vaclav Kotesovec_, May 24 2014

%F a(n) = I^(-n)*2^floor(n/2)*U(-n/2,1/2,-1), where U(a,b,x) is Kummer's hypergeometric function of the second kind. - _Peter Luschny_, May 24 2014

%F a(n) = numerator(a(n-1) + (a(n-2)/2)*(n-2)) for a(1)=0 and a(2)=1. - _Terry D. Grant_, May 04 2017

%p a := n -> I^(-n)*2^floor(n/2)*KummerU(-n/2,1/2,-1);

%p seq(round(evalf(a(n), 32)), n = 0..26); # _Peter Luschny_, May 24 2014

%t Table[Sum[StirlingS1[n, k] BellB[k]/2^(Ceiling[n/2] - k), {k, 0, n}], {n, 0, 100}]

%o (Maxima) makelist(sum(stirling1(n,k)*belln(k)/2^(ceiling(n/2)-k),k,0,n),n,0,41);

%Y Cf. A000085.

%K nonn

%O 0,3

%A _Emanuele Munarini_, May 23 2014