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A097397 Coefficients in asymptotic expansion of normal probability function.

%I A097397 #28 Nov 18 2023 08:27:27
%S A097397 1,1,1,5,9,129,57,9141,-36879,1430049,-15439407,418019205,-7404957255,
%T A097397 196896257505,-4656470025015,134136890777205,-3845524501226655,
%U A097397 123250625100419265,-4085349586734306015,145973136800663973765
%N A097397 Coefficients in asymptotic expansion of normal probability function.
%C A097397 a(0) + a(1)*x/(1-2*x) + a(2)*x^2/((1-2*x)*(1-4*x)) + ... = 1 + x + 3*x^2 + 15*x^3 + ...
%D A097397 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 932.
%H A097397 Vaclav Kotesovec, <a href="/A097397/b097397.txt">Table of n, a(n) for n = 0..200</a>
%H A097397 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A097397 E.g.f.: 1/sqrt(1 - log(1 + 2*x)). - _Seiichi Manyama_, Mar 05 2022
%F A097397 a(n) ~ n! * (-1)^(n+1) * 2^(n-1) / (log(n)^(3/2) * n) * (1 - 3*(gamma + 1)/(2*log(n)) + 15*(1 + 2*gamma + gamma^2 - Pi^2/6) / (8*log(n)^2)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Mar 05 2022
%F A097397 From _Seiichi Manyama_, Nov 18 2023: (Start)
%F A097397 a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * Stirling1(n,k).
%F A097397 a(0) = 1; a(n) = Sum_{k=1..n} (-2)^k * (1/2 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k). (End)
%t A097397 Table[Sum[2^(n - 2*k)*(2*k)!/k! * SeriesCoefficient[(1 - n + x)*Pochhammer[2 - n + x, -1 + n], {x, 0, k}], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 10 2019 *)
%o A097397 (PARI) a(n)=sum(k=0,n, 2^(n-2*k)*(2*k)!/k!* polcoeff(prod(i=0,n-1,x-i),k))
%o A097397 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-log(1+2*x)))) \\ _Seiichi Manyama_, Mar 05 2022
%Y A097397 Cf. A006252, A352070, A352073.
%K A097397 sign
%O A097397 0,4
%A A097397 _Michael Somos_, Aug 13 2004