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A385961 Decimal expansion of the value of the coefficient [x^3] Gamma(x).

%I A385961 #11 Jul 15 2025 05:45:49
%S A385961 9,8,1,7,2,8,0,8,6,8,3,4,4,0,0,1,8,7,3,3,6,3,8,0,2,9,4,0,2,1,8,5,0,8,
%T A385961 5,0,3,6,0,5,7,3,6,7,9,7,2,3,4,6,5,4,1,5,4,0,4,9,5,7,4,5,5,5,9,3,8,5,
%U A385961 6,8,3,9,2,4,8,6,9,3,4,5,0,9,4,1,0,5,9,7,7,0,5,1,8,7,5,7,0,6,5,9,5,5,8,8,5,0,6,7,0,4,3,6,8,2
%N A385961 Decimal expansion of the value of the coefficient [x^3] Gamma(x).
%C A385961 The Laurent series Gamma(x)  = 1/x + Sum_{i>=0} a_i x^i starts with a_0 = -gamma = -A001620, a_1 = A090998 . a_3 = 0.9817280868.. is here.
%H A385961 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) 8.321.1 gives recurrence.
%H A385961 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)
%F A385961 Equals (gamma^4 +6*gamma^2*zeta(2) +8*gamma*zeta(3) +3*zeta(2)^2 +6*zeta(4))/24 , gamma = A001620, zeta(2) = A013661, zeta(3)=A002117, zeta(4) = A013662.
%e A385961 0.981728086834400187336380294021850...
%p A385961 (gamma^4+6*gamma^2*Zeta(2)+8*gamma*Zeta(3)+3*Zeta(2)^2+6*Zeta(4))/24 ; evalf(%) ;
%Y A385961 Cf. A090998 [x^1], A385960 [x^2], A385962 [x^4].
%K A385961 nonn,cons
%O A385961 0,1
%A A385961 _R. J. Mathar_, Jul 13 2025