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Search: id:A163138
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| A163138 |
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G.f. satisfies: A(x) = exp( Sum_{n>=1} (2^n + A(x))^n * x^n/n ). |
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1
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| 1, 3, 20, 329, 22584, 7938470, 12605643936, 84977963809781, 2379247465188706528, 273419351336298753589802, 128009562526607810326874017088, 242979581192696030760182903464959706
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OFFSET
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0,2
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COMMENT
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More generally, we have the following identity:
If A(x,q) = exp( Sum_{n>=1} (q^n + A(x,q))^n * x^n/n ), then
A(x,q) = 1/(1-x*A(x,q))*exp( Sum_{n>=1} q^(n^2)/(1-q^n*x*A(x,q))^n*x^n/n ).
Conjecture: if q is an integer, then A(x,q) is a power series in x with integer coefficients.
Setting q=1 defines the g.f. of the large Schroeder numbers (A006318).
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FORMULA
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G.f.: A(x) = 1/(1-x*A(x))*exp( Sum_{n>=1} 2^(n^2)/(1 - 2^n*x*A(x))^n * x^n/n ).
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 20*x^2 + 329*x^3 + 22584*x^4 + 7938470*x^5 +...
log(A(x)) = [2 + A(x)]*x + [2^2 + A(x)]^2*x^2/2 + [2^3 + A(x)]^3*x^3/3 +...
log(A(x)*(1-xA(x))) = 2/(1-2xA(x))*x + 2^4/(1-4xA(x))^2*x^2/2 + 2^9/(1-8xA(x))^3*x^3/3 +...
log(A(x)) = 3*x + 31*x^2/2 + 834*x^3/3 + 86227*x^4/4 + 39339038*x^5/5 +...
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PROGRAM
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (2^m+A)^m*x^m/m)+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
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Sequence in context: A136551 A086229 A130531 this_sequence A003150 A138897 A089943
Adjacent sequences: A163135 A163136 A163137 this_sequence A163139 A163140 A163141
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 07 2009
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EXTENSIONS
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Comment corrected by Paul D. Hanna (pauldhanna(AT)juno.com), Aug 08 2009
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