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

%I

%S 1,1,4,21,164,1590,18984,266154,4306672,78850080,1612769040,

%T 36436534200,901265930784,24223557739056,702975780428544,

%U 21907246213656720,729670520457987840,25867686811627795200,972505009580975483904

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

%C A179496(n) = A179495(n+1)/(n+1). - _Vaclav Kotesovec_, Dec 25 2013

%F 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.

%F E.g.f. A(x) = G(x)/x where G(x) is the e.g.f. of A179495.

%F 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

%e E.g.f. A(x) = 1 + x + 4*x^2/2! + 21*x^3/3! + 164*x^4/4! + 1590*x^5/5! +...

%e ...

%e Define a triangular matrix where the e.g.f. of column k = A(x)^(k+1):

%e 1;

%e 1, 1;

%e 4/2!, 2, 1;

%e 21/3!, 10/2!, 3, 1;

%e 164/4!, 66/3!, 18/2!, 4, 1;

%e 1590/5!, 592/4!, 141/3!, 28/2!, 5, 1;

%e 18984/6!, 6500/5!, 1428/4!, 252/3!, 40/2!, 6, 1;

%e 266154/7!, 85548/6!, 17430/5!, 2840/4!, 405/3!, 54/2!, 7, 1;

%e ...

%e then the logarithm of the above matrix equals:

%e 0;

%e 1, 0;

%e 1, 2, 0;

%e 0, 2, 3, 0;

%e 0, 0, 3, 4, 0;

%e 0, 0, 0, 4, 5, 0;

%e 0, 0, 0, 0, 5, 6, 0; ...

%o (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)}

%Y Cf. A179495.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 25 2010

%E Name simplified by _Paul D. Hanna_, Jul 27 2010

%E Minor edits _Vaclav Kotesovec_, Mar 31 2014

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