A355603 - OEIS

%I #12 Jul 10 2022 06:32:15

%S 1,0,0,0,0,5,-15,70,-420,3024,-22050,202950,-2113650,24324300,

%T -305645340,4174483950,-61253992800,961049212200,-16054949350440,

%U 284505099278400,-5329752594075000,105239780964864000,-2184466455408699000,47550052231211237400

%N Expansion of e.g.f. (1 + x)^(x^4/24).

%H Seiichi Manyama, <a href="/A355603/b355603.txt">Table of n, a(n) for n = 0..451</a>

%F a(0) = 1; a(n) = -(n-1)!/24 * Sum_{k=5..n} (-1)^k * k/(k-4) * a(n-k)/(n-k)!.

%F a(n) = n! * Sum_{k=0..floor(n/5)} Stirling1(n-4*k,k)/(24^k * (n-4*k)!).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^(x^4/24)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/24*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)!/24*sum(j=5, i, (-1)^j*j/(j-4)*v[i-j+1]/(i-j)!)); v;

%o (PARI) a(n) = n!*sum(k=0, n\5, stirling(n-4*k, k, 1)/(24^k*(n-4*k)!));

%Y Cf. A007113, A355605.

%K sign

%O 0,6

%A _Seiichi Manyama_, Jul 09 2022