A179420 E.g.f. A(x) satisfies: A(A(x)) = x*A'(x) with A(0)=0, A'(0)=1.
0, 1, 2, 12, 132, 2200, 50280, 1482768, 54171376, 2381590944, 123292821600, 7390709937600, 506182300962624, 39180896544097152, 3396777800819754624, 327323946734658720000, 34831825328790915321600
Offset: 0
Examples
E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...
E.g.f. satisfies: A(A(x)) = x*A'(x) where:
A'(x) = 1 + 2*x + 12*x^2/2! + 132*x^3/3! + 2200*x^4/4! +...
A(A(x)) = x + 4*x^2/2! + 36*x^3/3! + 528*x^4/4! + 11000*x^5/5! +...
Related expansions begin:
A*Dx(A)/2! = 2*x^2/2! + 15*x^3/3! + 180*x^4/4! + 3150*x^5/5! +...
A*Dx(A*Dx(A))/3! = 6*x^3/3! + 104*x^4/4! + 2140*x^5/5! +...
A*Dx(A*Dx(A*Dx(A)))/4! = 24*x^4/4! + 770*x^5/5! + 24600*x^6/6! +...
A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 120*x^5/5! + 6264*x^6/6! +...
which generate iterations of A=A(x) as illustrated by:
A(A(x))/x = 1 + 2*A + 2^2*A*Dx(A)/2! + 2^3*A*Dx(A*Dx(A))/3! +...
A(A(A(x)))/x = 1 + 3*A + 3^2*A*Dx(A)/2! + 3^3*A*Dx(A*Dx(A))/3! +...
A_{-1}(x)/x = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! +-...(inverse).
Illustrate a main property of the iterations A_n(x) of A(x) by:
A(x) = A(A(x)) * A(x)/[x*d/dx A(x)];
A(x) = A_3(x) * A_2(x)/[x*d/dx A_2(x)];
A(x) = A_4(x) * A_3(x)/[x*d/dx A_3(x)]; ...
which can be shown consistent by the chain rule of differentiation.
...
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)/x 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)*A(x) for k>=0.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..150
Crossrefs
Programs
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Mathematica
a[n_] := a[n] = Module[{A}, A[x_] = x+x^2+Sum[a[m]*x^m/m!, {m, 3, n-1}]; If[n<3, n!*Coefficient[A[x], x, n], n!*Coefficient[A[A[x]], x, n]/(n-2)] ]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jan 15 2018, translated from PARI *) -
Maxima
Co(n, k, F):=if k=1 then F(n) else sum(F(i+1)*Co(n-i-1, k-1, F), i, 0, n-k); a(n):=if n=0 then 0 else if n<3 then 1 else sum(Co(n,k,a)*a(k),k,2,n-1)/(n-2); /* Vladimir Kruchinin, Jun 29 2011 */
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PARI
{a(n)=local(A=x+x^2+sum(m=3,n-1,a(m)*x^m/m!)+x*O(x^n));if(n<3,n!*polcoeff(A,n),n!*polcoeff(subst(A,x,A),n)/(n-2))}
Formula
E.g.f. A(x) equals the e.g.f. of column 0 in the matrix log of the Riordan array (A(x)/x, A(x)).
Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x,
then A=A(x) satisfies:
A(x)/x = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...
A_{-1}(x)/x = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...
A_n(x)/x = 1 + n*A + n^2*A*Dx(A)/2! + n^3*A*Dx(A*Dx(A))/3! + n^4*A*Dx(A*Dx(A*Dx(A)))/4! +...
where Dx(F) = d/dx(x*F).
Further, we have: A(x) = A_{n+1}(x) * A_n(x)/[x*d/dx A_n(x)] which holds for all n.
a(n)=sum(k=2..n-1, R(n-1,k-1)*a(k))/(n-2), n>2, a(1)=1, a(2)=1, where R is the Riordan array (A(x)/x, A(x)). [Vladimir Kruchinin, Jun 29 2011]
E.g.f. satisfies: A(x) = Series_Reversion(-G(-x)) where G(x) is the e.g.f. of A193202 and satisfies: G(G(x)) = x*G'(G(x)). [Paul D. Hanna, Jul 22 2011]