|
| |
|
|
A143138
|
|
E.g.f.: A(x) = x + (exp(A(x)) - 1)^2.
|
|
2
| |
|
|
1, 2, 18, 254, 5010, 126902, 3926538, 143539454, 6053432130, 289293272102, 15450565342938, 911991586990574, 58955877533817810, 4142488437549926102, 314346159031755778218, 25620077133245941688414
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| Radius of convergence is r = log((2+sqrt(3))/2)/2 - (2-sqrt(3))/2 = 0.17793076... where A(r) = log((sqrt(3)+1)/2) = 0.311905358...
|
|
|
FORMULA
| E.g.f.: A(x) = Series_Reversion( x - (exp(x) - 1)^2 ).
E.g.f. derivative: A'(x) = 1/(1 + 2*exp(A(x)) - 2*exp(2*A(x)) ).
a(n)=(n-1)!*(sum(k=0..n-1, binomial(n+k-1,n-1)*sum(j=0..k, (-1)^(j)*binomial(k,j)*sum(l=0..j, (binomial(j,l)*(2*(j-l))!*(-1)^(l-j)*stirling2(n-l+j-1,2*(j-l)))/(n-l+j-1)!)))), n>0. [From Vladimir Kruchinin, Feb 08 2012]
|
|
|
EXAMPLE
| A(x) = x + 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! +...
exp(A(x)) - 1 = G(x) = the g.f. of A143139:
G(x) = x + 3*x^2/2! + 25*x^3/3! + 351*x^4/4! + 6901*x^5/5! +...
G(x)^2 = 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! +...
|
|
|
PROG
| (PARI) {a(n)=local(A=x+O(x^n)); for(i=0, n, A=x + (exp(A)-1)^2); n!*polcoeff(A, n)}
(PARI) {a(n)=n!*polcoeff(serreverse(x-(exp(x+x*O(x^n))-1)^2), n)}
(Maxima) a(n):=(n-1)!*(sum(binomial(n+k-1, n-1)*sum((-1)^(j)*binomial(k, j)*sum((binomial(j, l)*(2*(j-l))!*(-1)^(l-j)*stirling2(n-l+j-1, 2*(j-l)))/(n-l+j-1)!, l, 0, j), j, 0, k), k, 0, n-1)); [From Vladimir Kruchinin, Feb 08 2012]
|
|
|
CROSSREFS
| Cf. A143139.
Sequence in context: A138437 A121429 A109517 * A151362 A099880 A141009
Adjacent sequences: A143135 A143136 A143137 * A143139 A143140 A143141
|
|
|
KEYWORD
| nonn,changed
|
|
|
AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jul 27 2008
|
| |
|
|
