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A224922
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O.g.f. satisfies: A(x) = Sum_{n>=0} A000110(n)*x^n*A(x)^n, where A000110 are the Bell numbers.
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4
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1, 1, 3, 12, 56, 288, 1586, 9201, 55675, 349159, 2260209, 15063260, 103201968, 726380164, 5252083746, 39029442336, 298340012448, 2348365289852, 19058365017840, 159659978176454, 1382148854813222, 12373696834781918, 114606230432935860, 1098199966940781258
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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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O.g.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1 - k*x*A(x)).
O.g.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - 1*x*A(x)/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - x*A(x)/(1 - 3*x*A(x)/(1 - x*A(x)/(1 - 4*x*A(x)/1 - ... ))))))))), a continued fraction.
a(n) = [x^n] ( Sum_{k>=0} A000110(k)*x^k )^(n+1) / (n+1).
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EXAMPLE
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O.g.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 288*x^5 + 1586*x^6 +...
The o.g.f. satisfies:
(1) A(x) = 1 + x*A(x) + 2*x^2*A(x)^2 + 5*x^3*A(x)^3 + 15*x^4*A(x)^4 + 52*x^5*A(x)^5 + 203*x^6*A(x)^6 +...+ A000110(n)*x^n*A(x)^n +...
(2) A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)^2/((1-x*A(x))*(1-2*x*A(x))) + x^3*A(x)^3/((1-x*A(x))*(1-2*x*A(x))*(1-3*x*A(x))) + x^4*A(x)^4/((1-x*A(x))*(1-2*x*A(x))*(1-3*x*A(x))*(1-4*x*A(x))) +...
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PROG
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(PARI) {a(n)=if(n<0, 0, polcoeff( 1/x*serreverse(x/serlaplace(exp(exp(x+x*O(x^n))-1))), n))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*A^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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