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A276371 Expansion of e.g.f. exp(x/2)/(2 - exp(2*x))^(1/4).

%I A276371 #21 Nov 16 2023 11:51:41
%S A276371 1,1,3,19,177,2161,32643,587539,12273537,291853441,7782998883,
%T A276371 230028553459,7462717994097,263654454838321,10075889406229923,
%U A276371 414147167601017779,18217983822073897857,853975145498805244801,42495107452208870429763,2237264405984004517212499,124243242448367338311920817,7258224393227482972980320881,444967879322677755285771182403,28563002475012109334240250609619
%N A276371 Expansion of e.g.f. exp(x/2)/(2 - exp(2*x))^(1/4).
%H A276371 Paul D. Hanna, <a href="/A276371/b276371.txt">Table of n, a(n) for n = 0..520</a>
%F A276371 E.g.f. A(x) satisfies: A'(x) = A(x)*(1 + A(x)^4)/2 with A(0)=1.
%F A276371 a(2*n) = 0 (mod 3), a(2*n+1) = 1 (mod 3), for n>=0.
%F A276371 a(n) ~ Gamma(3/4) * 2^n * n^(n-1/4) / (sqrt(Pi) * exp(n) * (log(2))^(n+1/4)). - _Vaclav Kotesovec_, Sep 11 2016
%F A276371 From _Seiichi Manyama_, Nov 16 2023: (Start)
%F A276371 a(n) = Sum_{k=0..n} (-2)^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling2(n,k).
%F A276371 a(0) = 1; a(n) = Sum_{k=1..n} (-2)^k * (3/2 * k/n - 2) * binomial(n,k) * a(n-k).
%F A276371 a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n-1} 2^k * binomial(n-1,k) * a(n-k). (End)
%e A276371 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 177*x^4/4! + 2161*x^5/5! + 32643*x^6/6! + 587539*x^7/7! + 12273537*x^8/8! + 291853441*x^9/9! + 7782998883*x^10/10! +...
%e A276371 such that A(x) = exp(x/2)/(2 - exp(2*x))^(1/4).
%t A276371 With[{nn = 50}, CoefficientList[Series[Exp[x/2]/(2 - Exp[2*x])^(1/4), {x, 0, nn}], x] Range[0, nn]!] (* _G. C. Greubel_, Apr 09 2017 *)
%o A276371 (PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); A = exp(X/2)/(2-exp(2*X))^(1/4); n!*polcoeff(A, n)}
%o A276371 for(n=0,30,print1(a(n),", "))
%Y A276371 Cf. A124212.
%K A276371 nonn
%O A276371 0,3
%A A276371 _Paul D. Hanna_, Sep 09 2016