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A193112
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G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(2*n+1).
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5
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1, 1, 3, 13, 63, 328, 1796, 10200, 59529, 354837, 2151079, 13221261, 82200739, 516053099, 3266812048, 20829635112, 133651716406, 862342656359, 5591505085491, 36416212224801, 238114435569354, 1562560513492974, 10287406857203911
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. A(x) satisfies the continued fraction:
1 = A(x)/(1+ x*A(x)^2/(1- x*(1+x)*A(x)^2/(1+ x^3*A(x)^2/(1+ x^2*(1-x^2)*A(x)^2/(1+ x^5*A(x)^2/(1- x^3*(1+x^3)*A(x)^2/(1+ x^7*A(x)^2/(1+ x^4*(1-x^4)*A(x)^2/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 63*x^4 + 328*x^5 + 1796*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^3 - x^3*A(x)^5 + x^6*A(x)^7 + x^10*A(x)^9 - x^15*A(x)^11 - x^21*A(x)^13 ++--...
Related expansions.
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 303*x^4 + 1662*x^5 + 9447*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 760*x^4 + 4401*x^5 +...
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PROG
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(PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(2*(#A))+1, (-x)^(m*(m+1)/2)*Ser(A)^(2*m+1)), #A-1)); if(n<0, 0, A[n+1])}
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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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