A355605 - OEIS

%I #8 Jul 09 2022 11:05:49

%S 1,0,0,3,-6,20,0,-126,1260,-4320,5040,180180,-2601720,31309200,

%T -372756384,4877195400,-70178799600,1099333347840,-18429818232960,

%U 327676010785200,-6146676161388000,121301442091851840,-2512746856371628800,54527094987619716000

%N Expansion of e.g.f. (1 + x)^(x^2/2).

%F a(0) = 1; a(n) = -(n-1)!/2 * Sum_{k=3..n} (-1)^k * k/(k-2) * a(n-k)/(n-k)!.

%F a(n) = n! * Sum_{k=0..floor(n/3)} Stirling1(n-2*k,k)/(2^k * (n-2*k)!).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^(x^2/2)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*log(1+x))))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!/2*sum(j=3, i, (-1)^j*j/(j-2)*v[i-j+1]/(i-j)!)); v;

%o (PARI) a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 1)/(2^k*(n-2*k)!));

%Y Cf. A007121, A355603.

%K sign

%O 0,4

%A _Seiichi Manyama_, Jul 09 2022