A375173 Expansion of e.g.f. exp( (1/(1 - 4*x)^(1/2) - 1)/2 ).
1, 1, 7, 79, 1225, 24121, 575311, 16105447, 517380529, 18752175505, 756760712311, 33645775575391, 1633792107752377, 86022043957561609, 4880923725657950335, 296882100064302393271, 19269430292162925519841, 1329278651404123963041697
Offset: 0
Links
- Eric Weisstein's World of Mathematics, Bell Polynomial.
Programs
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Mathematica
Table[4^n * Sum[Abs[StirlingS1[n, k]] * BellB[k, 1/2] / 2^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 02 2024 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-4*x)^(1/2)-1)/2)))
Formula
a(n) = Sum_{k=0..n} 4^(n-k) * |Stirling1(n,k)| * A004211(k) = 4^n * Sum_{k=0..n} (1/2)^k * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
From Vaclav Kotesovec, Aug 02 2024: (Start)
a(n) = 6*(2*n - 3)*a(n-1) - (48*n^2 - 192*n + 191)*a(n-2) + 32*(n-3)*(n-2)*(2*n - 5)*a(n-3).
a(n) ~ 2^(2*n - 1/6) * n^(n - 1/3) / (sqrt(3) * exp(n - 3*2^(-4/3)*n^(1/3) + 1/2)) * (1 - 31/(72*2^(2/3)*n^(1/3)) - 4607/(20736*2^(1/3)*n^(2/3))). (End)
a(n) = (1/exp(1/2)) * (-4)^n * n! * Sum_{k>=0} binomial(-k/2,n)/(2^k * k!). - Seiichi Manyama, Jan 18 2025
Comments