OFFSET
0,3
COMMENTS
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
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