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A229548
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E.g.f. satisfies: A(x) = x + A'(x) * Integral A(x) dx.
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1
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1, 1, 4, 32, 412, 7664, 193080, 6308664, 259111920, 13063094736, 793227096672, 57121999975104, 4815054009921024, 469833060580971072, 52555992910144441344, 6682566041846191689984, 958596887629217123483904, 154093408614272498448416256, 27591229509407916324655477248
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n-1} binomial(n,k-1)*a(n-k)*a(k) for n>1 with a(1)=1.
a(n) ~ c * (n!)^2 / (2^n * n^(1/3)), where c = 3.081214203431821156695905553610151693827575050546... - Vaclav Kotesovec, Feb 20 2014
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EXAMPLE
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E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 32*x^4/4! + 412*x^5/5! + 7664*x^6/6! +...
such that A(x) = x + A'(x)*B(x) where
A'(x) = 1 + x + 4*x^2/2! + 32*x^3/3! + 412*x^4/4! + 7664*x^5/5! +...
B(x) = x^2/2! + x^3/3! + 4*x^4/4! + 32*x^5/5! + 412*x^6/6! + 7664*x^7/7! +...
so that B(x) = Integral A(x) dx (here integration does not include constant term).
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MATHEMATICA
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a = ConstantArray[0, 20]; a[[1]]=1; Do[a[[n]] = Sum[Binomial[n, k-1]*a[[n-k]]*a[[k]], {k, 1, n-1}], {n, 2, 20}]; a (* Vaclav Kotesovec, Feb 19 2014 *)
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PROG
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(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+A'*intformal(A+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
(PARI) /* Recurrence: */
{a(n)=if(n==1, 1, sum(k=1, n-1, binomial(n, k-1)*a(n-k)*a(k)))}
for(n=1, 20, 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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