login
A179496
E.g.f. satisfies: A(x) = A(x)^2*(1 + x*A(x))/(1+x) - x*A'(x).
2
1, 1, 4, 21, 164, 1590, 18984, 266154, 4306672, 78850080, 1612769040, 36436534200, 901265930784, 24223557739056, 702975780428544, 21907246213656720, 729670520457987840, 25867686811627795200, 972505009580975483904
OFFSET
0,3
COMMENTS
A179496(n) = A179495(n+1)/(n+1). - Vaclav Kotesovec, Dec 25 2013
FORMULA
Define a triangular matrix where the e.g.f. of column k equals A(x)^(k+1), then the matrix log is the matrix L with L(n+1,n)=L(n+2,n)=n+1 and zeros elsewhere.
E.g.f. A(x) = G(x)/x where G(x) is the e.g.f. of A179495.
a(n) ~ sqrt(1+r) * n^n * r^n / exp(n), where r = -1-LambertW(-1, -exp(-2)) = 2.146193220620582585237... is the root of the equation log(1+r)=r-1. - Vaclav Kotesovec, Jan 04 2014
EXAMPLE
E.g.f. A(x) = 1 + x + 4*x^2/2! + 21*x^3/3! + 164*x^4/4! + 1590*x^5/5! +...
...
Define a triangular matrix where the e.g.f. of column k = A(x)^(k+1):
1;
1, 1;
4/2!, 2, 1;
21/3!, 10/2!, 3, 1;
164/4!, 66/3!, 18/2!, 4, 1;
1590/5!, 592/4!, 141/3!, 28/2!, 5, 1;
18984/6!, 6500/5!, 1428/4!, 252/3!, 40/2!, 6, 1;
266154/7!, 85548/6!, 17430/5!, 2840/4!, 405/3!, 54/2!, 7, 1;
...
then the logarithm of the above matrix equals:
0;
1, 0;
1, 2, 0;
0, 2, 3, 0;
0, 0, 3, 4, 0;
0, 0, 0, 4, 5, 0;
0, 0, 0, 0, 5, 6, 0; ...
PROG
(PARI) {a(n)=local(A=x+x^2+O(x^(n+1)), D=1); n!*polcoeff(1+sum(m=1, n+1, (D=A*deriv(x*D+O(x^(n+1))))/m!), n)}
CROSSREFS
Cf. A179495.
Sequence in context: A157503 A144010 A366184 * A339233 A107872 A008858
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 25 2010
EXTENSIONS
Name simplified by Paul D. Hanna, Jul 27 2010
Minor edits Vaclav Kotesovec, Mar 31 2014
STATUS
approved