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%I #18 Jan 03 2014 20:11:53
%S 1,1,3,21,241,3951,85499,2325205,76860673,3014656183,137784836475,
%T 7235668490589,431692029451009,28991550501283359,2174713803535479419,
%U 181001542259074421413,16618721538838243841185,1674634828088234390862727,184352162064651888588105243
%N E.g.f. satisfies: A(x) = Sum_{n>=0} ( Integral A(x)^n dx )^n.
%H Paul D. Hanna, <a href="/A234855/b234855.txt">Table of n, a(n) for n = 0..100</a>
%F E.g.f. satisfies: A'(x) = Sum_{n>=1} n * A(x)^n * ( Integral A(x)^n dx )^(n-1).
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 241*x^4/4! + 3951*x^5/5! +...
%e To illustrate how the terms are generated, form a table of coefficients of x^k/k!, k>=0, in (Integral A(x)^n dx)^n for n>=0 like so:
%e n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
%e n=1: [0, 1, 1, 3, 21, 241, 3951, 85499, 2325205, 76860673, ...];
%e n=2: [0, 0, 2, 12, 88, 920, 13328, 254744, 6161568, 182632584, ...];
%e n=3: [0, 0, 0, 6, 108, 1710, 29700, 600642, 14344092, 403670790, ...];
%e n=4: [0, 0, 0, 0, 24, 960, 28800, 826560, 24665088, 793449216, ...];
%e n=5: [0, 0, 0, 0, 0, 120, 9000, 462000, 20958000, 922005000, ...];
%e n=6: [0, 0, 0, 0, 0, 0, 720, 90720, 7378560, 504040320, ...];
%e n=7: [0, 0, 0, 0, 0, 0, 0, 5040, 987840, 120022560, ...];
%e n=8: [0, 0, 0, 0, 0, 0, 0, 0, 40320, 11612160, ...];
%e n=9: [0, 0, 0, 0, 0, 0, 0, 0, 0, 362880, ...]; ...
%e then the column sums form the terms of this sequence.
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,30,intformal( (A+x*O(x^n))^k )^k));n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A232552, A107595.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 01 2014
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