This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385965 #20 Aug 08 2025 16:35:49
%S A385965 0,4,2,0,0,2,6,3,5,0,3,4,0,9,5,2,3,5,5,2,9,0,0,3,9,3,4,8,7,5,4,2,9,8,
%T A385965 1,8,7,1,1,3,9,4,5,0,0,4,0,1,1,0,6,0,9,3,5,2,2,0,6,5,8,1,2,9,7,6,1,8,
%U A385965 0,0,9,6,8,7,5,9,7,5,9,8,8,5,4,7,1,0,7,7,0,1,2,9,4,7,8,7,7,1,3,2,3,3,5,3,2,0,0,0,2,2,2,0,0,0,0,1,8
%N A385965 Decimal expansion of the absolute value of the coefficient [x^4] 1/Gamma(x).
%C A385965 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_4 = -0.04200263... , absolute value here. Recurrence (i-1)*c_i = gamma *c_{i-1} - Sum_{k=2..i-1} (-1)^k*zeta(k) * c_{i-k} .
%H A385965 Paolo Xausa, <a href="/A385965/b385965.txt">Table of n, a(n) for n = 0..10000</a>
%H A385965 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 A385965 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 A385965 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 A385965 Simon Plouffe, <a href="https://plouffe.fr/simon/inverter.txt">Table up to c_15</a>, (2004)
%H A385965 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 A385965 Equals (-4*zeta(3) +Pi^2*gamma -2*gamma^3)/12, gamma = A001620, zeta(3) = A002117, Pi = A000796.
%e A385965 -0.042002635034095235529003934875....
%p A385965 (4*Zeta(3)-Pi^2*gamma+2*gamma^3)/12 ; evalf(%) ;
%t A385965 First[RealDigits[(Pi^2*EulerGamma - 2*EulerGamma^3 - 4*Zeta[3])/12, 10, 100, -1]] (* or *)
%t A385965 First[RealDigits[Module[{x}, SeriesCoefficient[1/Gamma[x], {x, 0, 4}]], 10, 100, -1]] (* _Paolo Xausa_, Aug 08 2025 *)
%Y A385965 Cf. A001620 [x^2], A070860 [x^3], A385966 [x^5].
%K A385965 nonn,cons
%O A385965 0,2
%A A385965 _R. J. Mathar_, Jul 13 2025