OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. satisfies: [x^n] A(x)^(n+1) = (n+1)^(n+1) for n>=0.
G.f. satisfies: A(x) = G(x/A(x)) where A(x*G(x)) = G(x) = Sum_{n>=0} (n+1)^n*x^n.
EXAMPLE
G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 231*x^4 + 2844*x^5 +...
G.f. satisfies A(x) = G(x/A(x)) where A(x*G(x)) = G(x) begins:
G(x) = 1 + 2*x + 3^2*x^2 + 4^3*x^3 + 5^4*x^4 + 6^5*x^5 + 7^6*x^6 +...
so that:
A(x) = 1 + 2*x/A(x) + 3^2*x^2/A(x)^2 + 4^3*x^3/A(x)^3 + 5^4*x^4/A(x)^4 +...
The coefficients in A(x)^n for n=1..8 begin:
A^1: [(1), 2, 5, 26, 231, 2844, 43854, 803578, ...];
A^2: [1, (4), 14, 72, 591, 6872, 102070, 1823024, ...];
A^3: [1, 6, (27), 146, 1140, 12546, 179105, 3112332, ...];
A^4: [1, 8, 44, (256),1954, 20488, 280848, 4740128, ...];
A^5: [1, 10, 65, 410,(3125),31512, 415020, 6793750, ...];
A^6: [1, 12, 90, 616, 4761,(46656), 591638, 9384288, ...];
A^7: [1, 14, 119, 882, 6986, 67214, (823543), 12652712, ...];
A^8: [1, 16, 152, 1216, 9940, 94768, 1126992, (16777216), ...]; ...
where the coefficient of x^n in A(x)^(n+1) equals (n+1)^(n+1).
PROG
(PARI) {a(n)=polcoeff(x/serreverse(x*sum(m=0, n+1, (m+1)^m*x^m)+x^2*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 22 2011
STATUS
approved