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A375173 Expansion of e.g.f. exp( (1/(1 - 4*x)^(1/2) - 1)/2 ).

%I A375173 #27 Feb 16 2025 08:34:07
%S A375173 1,1,7,79,1225,24121,575311,16105447,517380529,18752175505,
%T A375173 756760712311,33645775575391,1633792107752377,86022043957561609,
%U A375173 4880923725657950335,296882100064302393271,19269430292162925519841,1329278651404123963041697
%N A375173 Expansion of e.g.f. exp( (1/(1 - 4*x)^(1/2) - 1)/2 ).
%C A375173 For k >= 2, the difference a(n+k) - a(n) is divisible by k. It follows that for each k, the sequence formed by taking a(n) modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes [1, 1, 7, 9, 5, 1, 1, 7, 9, 5, ...], a purely periodic sequence of period 5. Cf. A047974. - _Peter Bala_, Feb 11 2025
%H A375173 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A375173 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.
%F A375173 From _Vaclav Kotesovec_, Aug 02 2024: (Start)
%F A375173 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).
%F A375173 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)
%F A375173 a(n) = (1/exp(1/2)) * (-4)^n * n! * Sum_{k>=0} binomial(-k/2,n)/(2^k * k!). - _Seiichi Manyama_, Jan 18 2025
%t A375173 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 *)
%o A375173 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-4*x)^(1/2)-1)/2)))
%Y A375173 Cf. A080893, A375175.
%Y A375173 Cf. A004211.
%K A375173 nonn,easy
%O A375173 0,3
%A A375173 _Seiichi Manyama_, Aug 02 2024