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A001028
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E.g.f. satisfies A'(x) = 1 + A(A(x)), A(0)=0.
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10
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1, 1, 2, 7, 37, 269, 2535, 29738, 421790, 7076459, 138061343, 3089950076, 78454715107, 2238947459974, 71253947372202, 2511742808382105, 97495087989736907, 4145502184671892500, 192200099033324115855, 9676409879981926733908, 527029533717566423156698
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OFFSET
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1,3
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COMMENTS
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The e.g.f. is diverging (see the Math Overflow link). - Pietro Majer, Jan 29 2017
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REFERENCES
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This functional equation (for f(x)=1+A(x-1)) was the subject of problem B5 of the 2010 Putnam exam.
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LINKS
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FORMULA
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E.g.f. satisfies: A(x) = Series_Reversion( Integral 1/(1 + A(x)) dx ). - Paul D. Hanna, Jun 27 2015
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MAPLE
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A:= proc(n) option remember; local T; if n=0 then 0 else T:= A(n-1); unapply(convert(series(Int(1+T(T(x)), x), x, n+1), polynom), x) fi end: a:= n-> coeff(A(n)(x), x, n)*n!: seq(a(n), n=1..22); # Alois P. Heinz, Aug 23 2008
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MATHEMATICA
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terms = 21; A[_] = 0; Do[A[x_] = x + Integrate[A[A[x]], x] + O[x]^(n+1) // Normal, {n, terms}];
Rest[CoefficientList[A[x], x]]*Range[terms]! (* Jean-François Alcover, Dec 07 2011, updated Jan 10 2018 *)
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PROG
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(Maxima) Co(n, k, a):= if k=1 then a(n) else sum(a(i+1)*Co(n-i-1, k-1, a), i, 0, n-k); a(n):= if n=1 then 1 else (1/n)*sum(Co(n-1, k, a)*a(k), k, 1, n-1); makelist(n!*a(n), n, 1, 7); /* Vladimir Kruchinin, Jun 30 2011 */
(PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse(intformal(1/(1+A) +x*O(x^n)))); n!*polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn,eigen,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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