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A354020 E.g.f.: Integral exp(-x*tanh(x)) / cosh(x) dx.

%I A354020 #10 May 21 2022 18:53:38
%S A354020 1,-3,37,-967,42761,-2840651,263226797,-32337165967,5074716055825,
%T A354020 -988900430732179,233997381305743925,-66026674478583179351,
%U A354020 21886277194644477922841,-8415820163884142193795547,3713857173311984258246440381,-1863494782835471106307678337311
%N A354020 E.g.f.: Integral exp(-x*tanh(x)) / cosh(x) dx.
%F A354020 E.g.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! may be defined as follows.
%F A354020 (1) A(x) = Integral exp(-x*tanh(x)) / cosh(x) dx.
%F A354020 (2) A(x) = lim_{N->oo} (x/N) * Sum_{n=0..N-1} cosh(n*x/N)^n / cosh((n+1)*x/N)^(n+1).
%e A354020 E.g.f.: A(x) = x - 3*x^3/3! + 37*x^5/5! - 967*x^7/7! + 42761*x^9/9! - 2840651*x^11/11! + 263226797*x^13/13! - 32337165967*x^15/15! + ...
%e A354020 such that d/dx A(x) = exp(-x*tanh(x)) / cosh(x).
%e A354020 The e.g.f. A(x) may be obtained as the limit of the finite series given by
%e A354020 A(x) = lim_{N->oo} (x/N) * [ 1/cosh(x/N) + cosh(x/N)/cosh(2*x/N)^2 + cosh(2*x/N)^2/cosh(3*x/N)^3 + cosh(3*x/N)^3/cosh(4*x/N)^4 + cosh(4*x/N)^4/cosh(5*x/N)^5 + ... + cosh((N-1)*x/N)^(N-1)/cosh(x)^N ].
%e A354020 Related limits.
%e A354020 The following limits yield the e.g.f. of this sequence, A(x).
%e A354020 A(x) = lim_{N->oo} (x/N) * Sum_{n=0..2*N-1} (1-x/N)^n * (1 + (1-x/N)^n )^n / (1 + (1-x/N)^(n+1) )^(n+1).
%e A354020 A(x) = lim_{N->oo} (x/N) * Sum_{n=0..2*N-1} exp(-n*x/N) * (1 + exp(-n*x/N) )^n / (1 + exp(-(n+1)*x/N) )^(n+1).
%e A354020 A(x) = lim_{N->oo} (2*x/N) * Sum_{n=0..N-1} exp(-2*n*x/N) * (1 + exp(-2*n*x/N) )^n / (1 + exp(-2*(n+1)*x/N) )^(n+1).
%e A354020 A(x) = lim_{N->oo} (x/N) * Sum_{n=0..N-1} cosh(n*x/N)^n / cosh((n+1)*x/N)^(n+1).
%e A354020 A(x) = lim_{N->oo} (x/N) * Sum_{n=0..N-1} cosh(n*x/N)^n / (cosh(x/N)*cosh(n*x/N) + sinh(x/N)*sinh(n*x/N))^(n+1).
%e A354020 A(x) = lim_{N->oo} (x/N) * Sum_{n=0..N-1} 1/((1 + tan(x/N)*tanh(n*x/N))^n * cosh(n*x/N)).
%e A354020 A(x) = lim_{N->oo} (x/N) * Sum_{n=0..N-1} 1/((1 + (x/N)*tanh(n*x/N))^n * cosh(n*x/N)).
%e A354020 A(x) = lim_{N->oo} (x/N) * Sum_{n=0..N-1} exp(-n*x/N * tanh(n*x/N)) / cosh(n*x/N).
%e A354020 The final limit is in the form of an integral, giving the formula
%e A354020 A(x) = Integral exp(-x*tanh(x)) / cosh(x) dx.
%o A354020 (PARI) {a(n) = my(A = intformal( exp(-x * tanh(x +O(x^(2*n+1))))/cosh(x +O(x^(2*n+1)) ) )); (2*n-1)!*polcoeff(A,2*n-1)}
%o A354020 for(n=1,20,print1(a(n),", "))
%K A354020 sign
%O A354020 1,2
%A A354020 _Paul D. Hanna_, May 18 2022