A190863 - OEIS

%I #14 Oct 23 2025 05:02:25

%S 1,1,1,4,13,16,61,736,1849,-12032,15481,1386496,737221,-139053056,

%T 8889973,22783492096,577232881,-4931048505344,9090479089,

%U 1391155770032128,747609240061,-492982133885763584,14533568173741,214285634326006595584,1446836117077033,-112041422722084502503424

%N Expansion of e.g.f. exp(x*sqrt(1+x^2)).

%H Vincenzo Librandi, <a href="/A190863/b190863.txt">Table of n, a(n) for n = 0..105</a>

%F a(n) = n!*(Sum_{k=1..n} (binomial(k/2,(n-k)/2)*((-1)^(n-k)+1))/k!)/2.

%t With[{nn=30},CoefficientList[Series[Exp[x Sqrt[1+x^2]],{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Mar 23 2013 *)

%o (Maxima)

%o a(n):=n!*sum((binomial((k)/2,(n-k)/2)*((-1)^(n-k)+1))/k!,k,1,n)/2;

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(x*sqrt(1+x^2)))) /* _Joerg Arndt_, May 22 2011 */

%K sign

%O 0,4

%A _Vladimir Kruchinin_, May 21 2011