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A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)*tanh(1)^k) / (2^n*n! * (1-tanh(1))^n).

%I A104053 #23 Feb 16 2025 08:32:56
%S A104053 0,1,0,1,-1,-1,-1,0,0,3,1,-5,18,-13,-7,-11,70,-135,65,-10,45,111,-609,
%T A104053 1215,-1350,1275,-621,-141,-1009,6188,-16758,27335,-26845,12474,-2548,
%U A104053 1883,10977,-81353,270004,-511791,584710,-420287,216468,-70169,-3599,-146691,1248210,-4715217,10303461,-14439411
%N A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)*tanh(1)^k) / (2^n*n! * (1-tanh(1))^n).
%C A104053 For n>0 the row sums = (-1)^(n-1) * (n-1)! For n odd, the sum of the absolute values of the coefficients in the n-th row = (2*(n-1))!/n! (every other entry of A001761).
%C A104053 The sum of the (2n)th du Bois-Reymond constants = 1/5 or is very close to 1/5.
%C A104053 For the 6th and 9th rows, the coefficients were adjusted from results of the residue evaluations so that double factorials ((2n)!! = 2^n*n! (A000165)) are in the denominators. For the 6th row they were multiplied by 3, for the 9th row they were multiplied by 9.
%C A104053 For n>1, Sum_{k=0..n} (n-k+1)*a(n,k) = (-1)^(n)*A001286(n-1) [A001286 are Lah numbers: (n-1)*n!/2].
%H A104053 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>.
%H A104053 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/duBois-ReymondConstants.html">du Bois-Reymond Constants</a>.
%F A104053 For n>1, C_2n = -3 - 2 * Residue_{x=i} (x^2/((1+x^2)^n * (tan(x) - x))) (see MathWorld article).
%F A104053 For n>1, Sum_{k=0..n} (-1)^(n+k)*a(n, k) = (2*(n-1))!/n! (i.e., A001761(n-1)).
%t A104053 Table[2 Residue[x^2/((1+x^2)^n (Tan[x]-x)), {x, I}], {n, 0, 9}]
%Y A104053 Cf. A000142, A000165, A001761, A062545, A062546, A085466, A085467.
%K A104053 hard,sign,tabl
%O A104053 0,10
%A A104053 _Gerald McGarvey_, Mar 02 2005
%E A104053 Added the keyword tabl _Gerald McGarvey_, Aug 20 2009