The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A179420 E.g.f. A(x) satisfies: A(A(x)) = x*A'(x) with A(0)=0, A'(0)=1. 17
0, 1, 2, 12, 132, 2200, 50280, 1482768, 54171376, 2381590944, 123292821600, 7390709937600, 506182300962624, 39180896544097152, 3396777800819754624, 327323946734658720000, 34831825328790915321600 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
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]
EXAMPLE
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.
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 *)
PROG
(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))}
(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 */
CROSSREFS
a(n)/n! = A221019(n)/A221020(n).
Sequence in context: A266489 A208830 A132472 * A346186 A080487 A077696
KEYWORD
eigen,nonn
AUTHOR
Paul D. Hanna, Jul 13 2010
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 17 10:05 EDT 2024. Contains 373445 sequences. (Running on oeis4.)