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E.g.f.: Sum_{n>=0} 2^(-n*(n+1)/2!) * Product_{k=0..n} tanh(2^k*x).
1

%I #7 Mar 30 2012 18:37:28

%S 1,2,4,-16,-704,-17728,-337616,11402624,3549329536,595083636992,

%T 71290751812864,-14356598757876736,-25888783158323201024,

%U -24314339173695033407488,-15893228305971746577453056,17044727383475526594464251904,160878265780790778493935871983616

%N E.g.f.: Sum_{n>=0} 2^(-n*(n+1)/2!) * Product_{k=0..n} tanh(2^k*x).

%C Signs of terms are of period 8: [+,+,+,-,-,-,-,+].

%F E.g.f.: Sum_{n>=0} sinh(x)^(n+1) * Product_{k=0..n} cosh(2^k*x)^(n-1-k).

%e E.g.f.: A(x) = x + 2*x^2/2! + 4*x^3/3! - 16*x^4/4! - 704*x^5/5! +...

%e where

%e A(x) = tanh(x) + tanh(x)*tanh(2*x)/2 + tanh(x)*tanh(2*x)*tanh(4*x)/2^3 + tanh(x)*tanh(2*x)*tanh(4*x)*tanh(8*x)/2^6 +...

%e A(x) = sinh(x)/cosh(x) + sinh(x)^2/cosh(2*x) + sinh(x)^3*cosh(x)/cosh(4*x) + sinh(x)^4*cosh(x)^2*cosh(2*x)/cosh(8*x) + sinh(x)^5*cosh(x)^3*cosh(2*x)^2*cosh(4*x)/cosh(16*x) + sinh(x)^6*cosh(x)^4*cosh(2*x)^3*cosh(4*x)^2*cosh(8*x)/cosh(32*x) +...

%o (PARI) {a(n)=local(A=sum(m=0,n,2^(-m*(m+1)/2!)*prod(k=0,m,tanh(2^k*x+x*O(x^n)))));n!*polcoeff(A,n)}

%o (PARI) {a(n)=local(X=x+x*O(x^n),A=sum(m=0,n,sinh(X)^(m+1)*prod(k=0,m,cosh(2^k*X)^(m-1-k))));n!*polcoeff(A,n)}

%Y Cf. A194456, A194026, A194027.

%K sign

%O 1,2

%A _Paul D. Hanna_, Aug 24 2011