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%I #8 Mar 21 2016 07:41:28
%S 1,3,6,34,120,1096,5040,56848,362880,5451136,39916800,688876288,
%T 6227020800,130789805056,1307674368000,29497569445888,355687428096000,
%U 9746045395173376,121645100408832000,3451902721622867968,51090942171709440000,1686006043164464644096
%N E.g.f.: Sum_{n>=1} log((1 + exp(2*x^n))/2).
%F a(2*n-1) = (2*n-1)!.
%e E.g.f.: A(x) = x + 3*x^2/2! + x^3 + 34*x^4/4! + x^5 + 1096*x^6/6! + x^7 + 56848*x^8/8! + x^9 + 5451136*x^10/10! + x^11 +...
%e where A(x) = log((1+exp(2*x))/2) + log((1+exp(2*x^2))/2) + log((1+exp(2*x^3))/2) + log((1+exp(2*x^4))/2) +...
%e The exponentiation of the e.g.f. begins:
%e exp(A(x)) = 1 + x + 4*x^2/2! + 16*x^3/3! + 104*x^4/4! + 696*x^5/5! + 6272*x^6/6! + 57856*x^7/7! + 652416*x^8/8! +...+ A203716(n)*x^n/n! +...
%t nmax = 25; Rest[Range[0, nmax]! * CoefficientList[Series[Sum[Log[1/(1 - Tanh[x^k])], {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Mar 21 2016 *)
%o (PARI) {a(n)=n!*polcoeff(sum(m=1,n,log((1+exp(2*x^m+x*O(x^n)))/2)),n)}
%Y Cf. A203709, A203716 (exp).
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 04 2012
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