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A179493 E.g.f. A(x) satisfies: L(x) = A(x)/(x*A'(x)) * L(A(x)) where L(x) = x + x*A(x). 1
0, 1, 2, 12, 108, 1420, 24660, 541968, 14547792, 465228720, 17385553440, 747776581200, 36566808933600, 2012537262763872, 123612631608883872, 8412289268206662720, 630378349868153698560, 51733701375836221013760 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..17.

FORMULA

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

Let A_n(x) denote the n-th iteration of e.g.f. A(x), then

. A_{n+1}(x) = -1 + (1 + A(x))*A_n'(x)*x^2/A_n(x)^2.

. L(x) = A_n(x)/(x*A_n'(x)) * L(A_n(x)) where L(x) = x + x*A(x).

...

Let L = L(x) = x + x*A(x), then:

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

. A_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).

EXAMPLE

E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1420*x^5/5! +...

Related expansions:

. L(x) = x + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 540*x^5/5! +...

. L(A(x)) = x + 4*x^2/2! + 30*x^3/3! + 348*x^4/4! + 5560*x^5/5! +...

. x*A'(x) = x + 4*x^2/2! + 36*x^3/3! + 432*x^4/4! + 7100*x^5/5! +...

. A(x)/x = 1 + x + 4*x^2/2! + 27*x^3/3! + 284*x^4/4! + 4110*x^5/5! +...

where L(x) = x + x*A(x) = A(x)/(x*A'(x)) * L(A(x)).

...

The RIORDAN ARRAY (A(x)/x, A(x)) begins:

1;

1, 1;

4/2!, 2, 1;

27/3!, 10/2!, 3, 1;

284/4!, 78/3!, 18/2!, 4, 1;

4110/5!, 880/4!, 159/3!, 28/2!, 5, 1;

77424/6!, 13220/5!, 1932/4!, 276/3!, 40/2!, 6, 1;

1818474/7!, 252828/6!, 30390/5!, 3608/4!, 435/3!, 54/2!, 7, 1; ...

where the g.f. of column k = A(x)^(k+1)/x^k for k>=0.

...

The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:

0;

1, 0;

2/2!, 2, 0;

6/3!, 4/2!, 3, 0;

48/4!, 12/3!, 6/2!, 4, 0;

540/5!, 96/4!, 18/3!, 8/2!, 5, 0;

8520/6!, 1080/5!, 144/4!, 24/3!, 10/2!, 6, 0;

172620/7!, 17040/6!, 1620/5!, 192/4!, 30/3!, 12/2!, 7, 0; ...

where the g.f. of column k = (k+1)*(x + x*A(x)) for k>=0.

...

To illustrate the inversion series, let L=L(x)=x + x*A(x), then:

. A(A(x)) = x + 4*x^2/2! + 36*x^3/3! + 480*x^4/4! + 8720*x^5/5! +...

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

. A_3(x) = x + 6*x^2/2! + 72*x^3/3! + 1260*x^4/4! + 29340*x^5/5! +...

. A_3(x)/x = 1 + 3*L + 3^2*L*Dx(L)/2! + 3^3*L*Dx(L*Dx(L))/3! +...

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

PROG

(PARI) {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); G=x*Ser(A); A[ #A]=polcoeff(1+subst(G, x, G)+O(x^#A)-(1+G)*deriv(G)*x^2/G^2, #A-1)/(#A-2)); if(n<1, 0, n!*A[n])}

CROSSREFS

Cf. A179494, A179420, A179421.

Sequence in context: A218652 A194786 A339301 * A193268 A235601 A007724

Adjacent sequences: A179490 A179491 A179492 * A179494 A179495 A179496

KEYWORD

eigen,nonn

AUTHOR

Paul D. Hanna, Jul 23 2010

EXTENSIONS

Typos in formula and example corrected by Paul D. Hanna, Jul 28 2010

STATUS

approved

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Last modified December 5 12:42 EST 2022. Contains 358586 sequences. (Running on oeis4.)