A234855 E.g.f. satisfies: A(x) = Sum_{n>=0} ( Integral A(x)^n dx )^n.

1, 1, 3, 21, 241, 3951, 85499, 2325205, 76860673, 3014656183, 137784836475, 7235668490589, 431692029451009, 28991550501283359, 2174713803535479419, 181001542259074421413, 16618721538838243841185, 1674634828088234390862727, 184352162064651888588105243

Offset: 0

Author

Paul D. Hanna, Jan 01 2014

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 241*x^4/4! + 3951*x^5/5! +...
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:
n=0: [1, 0, 0,  0,   0,    0,     0,      0,        0,         0, ...];
n=1: [0, 1, 1,  3,  21,  241,  3951,  85499,  2325205,  76860673, ...];
n=2: [0, 0, 2, 12,  88,  920, 13328, 254744,  6161568, 182632584, ...];
n=3: [0, 0, 0,  6, 108, 1710, 29700, 600642, 14344092, 403670790, ...];
n=4: [0, 0, 0,  0,  24,  960, 28800, 826560, 24665088, 793449216, ...];
n=5: [0, 0, 0,  0,   0,  120,  9000, 462000, 20958000, 922005000, ...];
n=6: [0, 0, 0,  0,   0,    0,   720,  90720,  7378560, 504040320, ...];
n=7: [0, 0, 0,  0,   0,    0,     0,   5040,   987840, 120022560, ...];
n=8: [0, 0, 0,  0,   0,    0,     0,      0,    40320,  11612160, ...];
n=9: [0, 0, 0,  0,   0,    0,     0,      0,        0,    362880, ...]; ...
then the column sums form the terms of this sequence.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A232552, A107595.

Programs

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)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. satisfies: A'(x) = Sum_{n>=1} n * A(x)^n * ( Integral A(x)^n dx )^(n-1).