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A385966 Decimal expansion of the value of the coefficient [x^5] 1/Gamma(x).

%I A385966 #21 Aug 08 2025 16:36:04
%S A385966 1,6,6,5,3,8,6,1,1,3,8,2,2,9,1,4,8,9,5,0,1,7,0,0,7,9,5,1,0,2,1,0,5,2,
%T A385966 3,5,7,1,7,7,8,1,5,0,2,2,4,7,1,7,4,3,4,0,5,7,0,4,6,8,9,0,3,1,7,8,9,9,
%U A385966 3,8,6,6,0,5,6,4,7,4,2,4,8,3,1,9,4,7,1,9,1,4,6,5,8,0,4,1,6,2,6,6,2,3,9,5,5,9,3,4,0,5,1,2,8
%N A385966 Decimal expansion of the value of the coefficient [x^5] 1/Gamma(x).
%C A385966 The Taylor series 1/Gamma(x)  = Sum_{i>=1} c_i x^i starts with c_1 = 1, c_2 = gamma = A001620, c_3 = -0.655878... = -A070860 . c_5 = 0.166538... here.
%H A385966 Paolo Xausa, <a href="/A385966/b385966.txt">Table of n, a(n) for n = 0..10000</a>
%H A385966 M. Abramowitz, I. A. Stegun, <a href="https://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, 6.1.34.
%H A385966 I. S. Gradsteyn, I. M. Ryzhik, <a href="https://doi.org/10.1016/C2010-0-64839-5">Tables of Series and Products</a>, Academic Press (2014) 8.321.2 gives recurrence.
%H A385966 R. J. Mathar, <a href="https://vixra.org/abs/2507.0094">Erratum to Exercise A4.2 in "An Introduction to the Theory of the Riemann Zeta Function"</a>, viXra:2507.0094 (2025)
%H A385966 Simon Plouffe, <a href="https://plouffe.fr/simon/inverter.txt">Table up to c_15</a>, (2004)
%H A385966 J. W. Wrench, <a href="https://doi.org/10.1090/S0025-5718-1968-0237078-4">Concerning two series for the Gamma Function</a>, Math. Comp. 22 (1968) 617-626, Table 5.
%F A385966 Equals (Pi^4 -60*Pi^2*gamma^2 +60*gamma^4 +480*gamma*zeta(3))/1440, gamma = A001620, zeta(3) = A002117, Pi = A000796.
%e A385966 0.16653861138229148950170079510210523571...
%p A385966 (Pi^4-60*Pi^2*gamma^2+60*gamma^4+480*gamma*Zeta(3))/1440 ; evalf(%) ;
%t A385966 First[RealDigits[(Pi^4 - 60*Pi^2*#^2 + 60*#^4 + 480*#*Zeta[3])/1440 & [EulerGamma], 10, 100]] (* or *)
%t A385966 First[RealDigits[Module[{x}, SeriesCoefficient[1/Gamma[x], {x, 0, 5}]], 10, 100]] (* _Paolo Xausa_, Aug 08 2025 *)
%Y A385966 Cf. A001620 [x^2], A070860 [x^3], A385965 [x^4].
%K A385966 nonn,cons
%O A385966 0,2
%A A385966 _R. J. Mathar_, Jul 13 2025