A384343 Expansion of Product_{k>=1} (1 + k*x)^((1/2)^(k+1)).
1, 1, -1, 3, -14, 86, -650, 5822, -60287, 708873, -9334633, 136142011, -2179136696, 37987580268, -716513806824, 14540745561432, -315936103907094, 7318039354370826, -180020739049731594, 4687207255550122014, -128782014195949550724, 3723598212075752653284, -113023054997369519314572
Offset: 0
Keywords
Programs
-
Mathematica
terms = 25; A[] = 1; Do[A[x] = -A[x] + 2*((1 + x)*A[x/(1 + x)])^(1/2) + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 29 2025 *)
-
PARI
my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, j!*stirling(k, j, 2))*x^k/k)))
Formula
G.f. A(x) satisfies A(x) = (1+x)^(1/2) * A(x/(1+x))^(1/2).
G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A000670(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A084784.
a(n) ~ (-1)^(n+1) * (n-1)! / (2*log(2)^(n+1)). - Vaclav Kotesovec, May 29 2025