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A205566
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G.f.: A(x) = 1 + x*A(x)*A(-x) + x^2*exp( Sum_{n>=1} 2*L(n)^2*x^(2*n)/n ), where A(x) = exp(Sum_{n>=1} L(n)*x^n/n).
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1
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1, 1, 1, 1, 2, 3, 3, 3, 4, 6, 17, 27, 44, 63, 71, 97, 98, 114, 203, 288, 498, 773, 3138, 5863, 8018, 10406, 19726, 32753, 40372, 47679, 89898, 154289, 199094, 277526, 439695, 742508, 880786, 1099183, 1765472, 2706524, 3378956, 4461436, 9651388, 16828143, 24527676
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OFFSET
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0,5
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COMMENTS
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Note: A(x)*A(-x) = exp( Sum_{n>=1} L(2*n)*x^(2*n)/n ) is a bisection of the g.f. A(x), where L(n) is the n-th coefficient in the logarithmic derivative of A(x).
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LINKS
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8 +...
Related expansions:
A(x)*A(-x) = 1 + x^2 + 3*x^4 + 3*x^6 + 6*x^8 + 27*x^10 + 63*x^12 + 97*x^14 + 114*x^16 + 288*x^18 + 773*x^20 + 5863*x^22 + 10406*x^24 +...
exp(Sum_{n>=1} 2*L(n)^2*x^(2*n)/n) = 1 + 2*x^2 + 3*x^4 + 4*x^6 + 17*x^8 + 44*x^10 + 71*x^12 + 98*x^14 + 203*x^16 + 498*x^18 + 3138*x^20 + 8018*x^22 +...
log(A(x)) = x + x^2/2 + x^3/3 + 5*x^4/4 + 6*x^5/5 + x^6/6 + x^7/7 + 5*x^8/8 + 10*x^9/9 + 106*x^10/10 + 111*x^11/11 +...+ L(n)*x^n/n +...
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PROG
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(PARI) {a(n)=local(L=vector(n+1, i, 1), A=Ser(L)); for(i=1, n, A=1+x*A*subst(A, x, -x+x*O(x^n))+x^2*exp(2*sum(m=1, #L\2, x^(2*m)*L[m]^2/m)+x*O(x^n)); L=Vec(deriv(log(A)))); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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