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A363307
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Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} (-x)^n * A(x)^n / Product_{k=1..n+1} (1 + (-x)^k).
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1
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1, 1, 2, 3, 5, 9, 18, 37, 78, 168, 367, 809, 1800, 4040, 9136, 20798, 47627, 109630, 253506, 588602, 1371688, 3207309, 7522302, 17691875, 41716878, 98600132, 233557375, 554358994, 1318282031, 3140435548, 7493519010, 17908158678, 42859216345, 102713690586, 246472881089
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OFFSET
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0,3
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COMMENTS
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A related identity: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^n / Product_{k=1..n+1} (1 + (-x)^k*A(x) ), which holds formally for all A(x).
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) 1 = Sum_{n>=0} (-x)^n * A(x)^n / Product_{k=1..n+1} (1 + (-x)^k).
(2) 1 = Sum_{n>=0} x^n / Product_{k=1..n+1} (1 - (-x)^k*A(x)).
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 18*x^6 + 37*x^7 + 78*x^8 + 168*x^9 + 367*x^10 + 809*x^11 + 1800*x^12 + ...
where
1 = 1/(1 - x) - x*A(x)/((1 - x)*(1 + x^2)) + x^2*A(x)^2/((1 - x)*(1 + x^2)*(1 - x^3)) - x^3*A(x)^3/((1 - x)*(1 + x^2)*(1 - x^3)*(1 + x^4)) + x^4*A(x)^4/((1 - x)*(1 + x^2)*(1 - x^3)*(1 + x^4)*(1 - x^5)) + ...
also,
1 = 1/(1 + x*A(x)) + x/((1 + x*A(x))*(1 - x^2*A(x))) + x^2/((1 + x*A(x))*(1 - x^2*A(x))*(1 + x^3*A(x))) + x^3/((1 + x*A(x))*(1 - x^2*A(x))*(1 + x^3*A(x))*(1 - x^4*A(x))) + x^4/((1 + x*A(x))*(1 - x^2*A(x))*(1 + x^3*A(x))*(1 - x^4*A(x))*(1 + x^5*A(x))) + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-1 + sum(n=0, #A, (-x)^n*Ser(A)^n / prod(k=1, n+1, (1 + (-x)^k ) ) ), #A); ); A[n+1]}
for(n=0, 35, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-1 + sum(n=0, #A, x^n / prod(k=1, n+1, (1 - (-x)^k*Ser(A)) ) ), #A); ); A[n+1]}
for(n=0, 35, print1(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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