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%I #12 Mar 21 2016 07:17:52
%S 1,1,4,16,104,696,6272,57856,652416,7657600,104244992,1475430144,
%T 23426373632,387521615872,7034561925120,132850810138624,
%U 2709375373672448,57456525327335424,1301169515685085184,30573796812553584640,760486440376336908288,19600568102376899608576
%N E.g.f.: Product_{n>=1} (exp(2*x^n) + 1)/2.
%H Vaclav Kotesovec, <a href="/A203716/b203716.txt">Table of n, a(n) for n = 0..350</a>
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 104*x^4/4! + 696*x^5/5! +...
%e where the e.g.f. equals the product:
%e A(x) = (exp(2*x)+1)/2 * (exp(2*x^2)+1)/2 * (exp(2*x^3)+1)/2 * (exp(2*x^4)+1)/2 *...
%e The log of the e.g.f. begins:
%e log(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 +...+ A203715(n)*x^n/n! +...
%e Note that the coefficients of the odd powers of x in log(A(x)) equals 1.
%t nmax = 25; Range[0, nmax]! * CoefficientList[Series[Product[1/(1 - Tanh[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 21 2016 *)
%o (PARI) {a(n)=n!*polcoeff(prod(k=1, n, (exp(2*x^k+x*O(x^n))+1)/2), n)}
%Y Cf. A203715 (log), A203709, A270664, A270665, A270666.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 04 2012
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