A179496 E.g.f. satisfies: A(x) = A(x)^2*(1 + x*A(x))/(1+x) - x*A'(x).

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

Links

Table of n, a(n) for n=0..18.

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

Adjacent sequences: A179493 A179494 A179495 * A179497 A179498 A179499

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