A380050 - OEIS

%I #11 Jan 11 2025 10:26:32

%S 1,1,3,9,25,25,-429,-4151,-8175,320625,5241475,23329801,-705579159,

%T -18521117303,-150119840493,3366485315145,138253031778721,

%U 1780881865542625,-28047359274759549,-1854674541474191351,-34985197604145203655,332608115115937927161

%N E.g.f. A(x) satisfies A(x) = sqrt( 1 + 2*x*exp(x)*A(x) ).

%F E.g.f.: exp( arcsinh(x*exp(x)) ).

%F E.g.f.: x*exp(x) + sqrt(1 + x^2*exp(2*x)).

%F a(n) = n! * Sum_{k=0..n} 2^k * k^(n-k) * binomial(k/2+1/2,k)/( (k+1)*(n-k)! ).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(asinh(x*exp(x)))))

%o (PARI) a(n) = n!*sum(k=0, n, 2^k*k^(n-k)*binomial(k/2+1/2, k)/((k+1)*(n-k)!));

%Y Cf. A006153, A380051.

%Y Cf. A297010, A380044.

%K sign

%O 0,3

%A _Seiichi Manyama_, Jan 11 2025