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A245378
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G.f. satisfies: A(x) = Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - (n+1)*x*A(x)) ).
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1
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1, 1, 3, 10, 39, 165, 743, 3507, 17199, 87126, 454159, 2430031, 13326623, 74856230, 430628069, 2538270783, 15343363603, 95233568052, 607850790015, 3996223189468, 27105153736781, 189947851239185, 1376864409041417, 10330672337146804, 80248762443834399, 645206035074873681
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OFFSET
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0,3
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COMMENTS
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Compare g.f. to an identity for C(x) = 1 + x*C(x)^2, the Catalan function:
C(x) = Sum_{n>=0} n! * x^n * C(x)^n / Product_{k=1..n} (1 + k*x*C(x)).
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LINKS
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FORMULA
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G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^n * (1+x)^n / Product_{k=1..n} (1 + k*x*A(x)).
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 165*x^5 + 743*x^6 +...
where we have the following identity:
A(x) = 1/((1+x)*(1-x*A(x))) + x/((1+x)^2*(1-2*x*A(x))) + x^2/((1+x)^3*(1-3*x*A(x)))+ x^3/((1+x)^4*(1-4*x*A(x)))+ x^4/((1+x)^5*(1-5*x*A(x))) + x^5/((1+x)^6*(1-6*x*A(x))) +...
is equal to
A(x) = 1 + x*A(x)*(1+x)/(1+x*A(x)) + 2!*x^2*A(x)^2*(1+x)^2/((1+x*A(x))*(1+2*x*A(x))) + 3!*x^3*A(x)^3*(1+x)^3/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))) + 4!*x^4*A(x)^4*(1+x)^4/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))) + 5!*x^5*A(x)^5*(1+x)^5/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))*(1+5*x*A(x))) +...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/((1+x)^(m+1)*(1 - (m+1)*x*A +x*O(x^n))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m!*x^m*A^m*(1+x)^m/prod(k=1, m, 1+k*x*A +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, 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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