A179421 E.g.f. A(x) satisfies: x*A(x) equals column 0 in the matrix log of the Riordan array (A(x), x*A(x)).

1, 1, 4, 33, 440, 8380, 211824, 6771422, 264621216, 12329282160, 671882721600, 42181858413552, 3013915118776704, 242626985772839616, 21821596448977248000, 2176989083049432207600, 239420370429753669425664

Offset: 0

Author

Paul D. Hanna, Jul 15 2010

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 440*x^4/4! +...
x*A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...
...
The RIORDAN ARRAY (A(x), x*A(x)) begins:
. 1;
. 1, 1;
. 4/2!, 2, 1;
. 33/3!, 10/2!, 3, 1;
. 440/4!, 90/3!, 18/2!, 4, 1;
. 8380/5!, 1240/4!, 177/3!, 28/2!, 5, 1;
. 211824/6!, 23800/5!, 2544/4!, 300/3!, 40/2!, 6, 1;
. 6771422/7!, 598788/6!, 49680/5!, 4520/4!, 465/3!, 54/2!, 7, 1; ...
where the e.g.f. of column k = A(x)^(k+1) for k>=0.
...
The MATRIX LOG of the above Riordan array (A(x), x*A(x)) begins:
. 0;
. 1, 0;
. 2/2!, 2, 0;
. 12/3!, 4/2!, 3, 0;
. 132/4!, 24/3!, 6/2!, 4, 0;
. 2200/5!, 264/4!, 36/3!, 8/2!, 5, 0;
. 50280/6!, 4400/5!, 396/4!, 48/3!, 10/2!, 6, 0;
. 1482768/7!, 100560/6!, 6600/5!, 528/4!, 60/3!, 12/2!, 7, 0; ...
where the e.g.f. of column k = (k+1)*x*A(x) for k>=0.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A179420, A179422, A179423, A179424.

Programs

PARI

{a(n)=local(A=1+x+sum(m=2,n-1,a(m)*x^m/m!)+x*O(x^n));if(n<2,n!*polcoeff(A,n),n!*polcoeff(subst(x*A,x,x*A)/x,n)/(n-1))}

Formula

a(n) = A179420(n+1)/(n+1).

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

Let L_{n}(x) denote the n-th iteration of L=L(x)=x*A(x) then:

. L(L(x)) = x*L'(x),

. L_{n+1}(x) = L(x)*x*[d/dx L_n(x)]/L_n(x),

. L(x)/x = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +...

. L_{n}(x)/x = 1 + n*L + n^2*L*Dx(L)/2! + n^3*L*Dx(L*Dx(L))/3! + n^4*L*Dx(L*Dx(L*Dx(L)))/4! +...

where Dx(F) = d/dx(x*F).